Algebra 8

ALGEBRA 8

For NAKHICHEVAN PRIVATE TURKISH HIGH SCHOOLS

ALGEBRA 8th CLASS For NAKHICHEVAN PRIVATE TURKISH HIGH SCHOOLS

CHAPTER I

Rational numbers

CHAPTER II

Radicals

CHAPTER III

Quadratic equations

CHAPTER IV

Inequalities

CHAPTER V

Powers with integer exponent

MEHMET AKİF ALTUNDAL MAAKIF PUBLICATIONS SHERUR 2002 3

CHAPTER I RATIONAL NUMBERS..................................................... 7 RATIONAL NUMBERS AND PROPERTIES ..................................................... 7 1. RATIONAL EXPRESSIONS ....................................................................... 7 2. SIMPLIFYING RATIONAL FRACTIONS ..................................................... 8 SUM AND DIFFERENCE OF FRACTIONS ....................................................... 8 3. ADDITION AND SUBTRACTION OF FRACTIONS WITH SAME DENOMINATOR .............................................................................................. 8 4. ADDITION AND SUBTRACTION OF FRACTIONS WITH DIFFERENT FRACTIONS .................................................................................................... 9 PRODUCT AND QUOTIENT OF FRACTIONS ................................................ 10 5. MULTIPLICATION AND POWER OF FRACTIONS..................................... 10 6. DIVISION OF FRACTIONS ...................................................................... 11 7. EVALUATION RATIONAL EXPRESSIONS ............................................... 12 CHAPTER II RADICALS....................................................................... 13 REAL NUMBERS ......................................................................................... 14 9. RATIONAL NUMBERS ........................................................................... 14 10. IRRATIONAL NUMBERS ...................................................................... 15 RADICALS................................................................................................... 15 11. SQUARE ROOTS .................................................................................. 15 12. 13.

x2 = a

EQUATION........................................................................... 16 CALCULATING VALUE OF SQUARE ROOT APPROXIMATELY .............. 16

14. y = x AND ITS GRAPH ................................................................... 17 PROPERTIES OF SQUARE ROOT ................................................................. 17 15. SQUARE ROOT OF PRODUCT AND FRACTION...................................... 17 16. SQUARE ROOT OF POWER .................................................................. 18 APPLICATION OF SQUARE ROOT............................................................... 19 17. MOVING THE FACTOR TO INSIDE OR OUTSIDE OF ROOT .................... 19 18. EVALUATION OF EXPRESSIONS WITH SQUARE ROOTS ....................... 19 CHAPTER III QUADRATIC EQUATIONS ........................................ 20 QUADRATIC EQUATION AND ITS ROOTS ................................................... 21

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DEFINITION OF QUADRATIC EQUATION AND INCOMPLETE QUADRATIC EQUATION ................................................................................................... 21 20. SOLVING QUADRATIC EQUATIONS WITH USING SQUARE OF BINOMIALS .................................................................................................. 22 FORMULA OF QUADRATIC EQUATION ...................................................... 24 21. SOLVING QUADRATIC EQUATION WITH FORMULA ............................ 24 22. SOLVING PROBLEMS WITH THE HELP OF QUADRATIC EQUATION ...... 25 23. VIET THEOREM .................................................................................. 25 RATIONAL EQUATIONS .............................................................................. 26 24. SOLVING RATIONAL EQUATIONS ....................................................... 26 25. SOLVING WORD PROBLEMS WITH RATIONAL EQUATIONS..................... 27 EXTRA EXERCISES FOR CHAPTER III ........................................................... 29 CHAPTER IV INEQUALITIES ............................................................. 33 NUMERICAL INEQUALITIES ...................................................................... 33 EXERCISES OF NUMERICAL INEQUALITIES ................................................. 34 28. PROPERTIES OF NUMERICAL INEQUALITIES ....................................... 36 EXERCISES OF PROPERTIES OF NUMERICAL INEQUALITIES ........................ 38 29. ADDITION AND MULTIPLICATION OF NUMERICAL INEQUALITIES ...... 40 EXERCISES OF ADDITION AND MULTİPLİCATİON OF NUMERICAL INEQUALITIES .............................................................................................. 42 INEQUALITIES WITH ONE VARIABLE AND SYSTEM OF INEQUALITIES .... 44 30. INTERVALS ........................................................................................ 44 EXERCISES OF INTERVALS .......................................................................... 44 31. SOLVING INEQUALITIES WITH ONE VARIABLE ................................... 46 EXERCISES OF SOLVING INEQUALITIES WITH ONE VARIABLE .................... 47 32. SOLVING SYSTEM OF INEQUALITIES WITH ONE VARIABLE ................ 52 EXERCISES OF SOLVING SYSTEM OF INEQUALITIES WITH ONE VARIABLE .. 55 EXTRA EXERCISES FROM CHAPTER IV........................................................ 59 CHAPTER V POWERS WITH INTEGER EXPONENT.................... 65 POWER WITH INTEGER EXPONENT AND PROPERTIES ............................. 65 33. DEFINITION OF POWER WITH NEGATIVE INTEGER ............................. 65 EXERCISES OF DEFINITION OF POWER WITH NEGATIVE INTEGER ............... 66 34. PROPERTIES OF EXPONENTIALS WITH INTEGER POWERS ................... 69 EXERCISES OF PROPERTIES OF EXPONENTIALS WITH INTEGER POWERS .... 69

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35. SCIENTIFIC NOTATION ....................................................................... 73 ERROR CALCULATIONS ............................................................................. 73 36. WRITING APPROXIMATELY (WITH ERROR) ........................................ 73 37. OPERATIONS ON APPROXIMATE VALUE ......................................... 74 EXTRA EXERCISES FOR CHAPTER V ............................................................ 75

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CHAPTER I Rational numbers CONTENT Rational numbers and properties Sum and difference of fractions Product and quotient of fractions

Rational numbers and properties 1. Rational expressions Expressions with numbers, variables and operations are algebraic expressions. Rational expressions are all algebraic expressions with fractions. Polynomials are algebraic expressions without fractions. Example: 7a 3b , m3 + n3 , ( x − y )( x 2 + y 2 ) are polynomials. x− y b n 5 , − 2 Example: 4a − , 2 are rational 2 2a + 1 x − 3xy + y 3 n + 1 expressions. Domain is the values of variables in rational expressions. These values don’t make undefined. 1 Example: 10 + is a rational expression. Domain is a ≠ 0 . a x Example: x + is a rational expression. Domain is x ≠ y . x− y 5 Example: is a rational expression. Domain is a ≠ 0 and a(a − 9) a ≠ 9. 3a − b 2 Example: Find value of if a = and b = −1,5 2ab 3 2 3 × − (−1,5) 3,5 3a − b 2 + 1,5 3 = = = = −1, 75 2 2 2ab − 2 2 × × (−1,5) × (−3) 3 3 7

Exercises

2. Simplifying rational fractions Fundamental property of fractions; If a, b, c are integers and b ≠ 0 , c ≠ 0 then a ac = b bc 21y Example: Simplify 3y2 21y 3 × y × 7 7 = = 3y2 3 × y × y y a2 − 9 ab + 3b 2 a −9 a 2 − 32 (a + 3)(a − 3) a − 3 = = = ab + 3b b(a + 3) b(a + 3) b 2x to 35y 3 Example: Change denominator of 7y Example: Simplify

2 x 2 x × 5 y 2 10 xy 2 = = 7 y 7 y × 5 y 2 35 y 3 5 to x − 2 y . Example: Change denominator of 2y − x 5 × (−1) −5 −5 = = (2 y − x) × (−1) −2 y + x x − 2 y Exercises

Sum and difference of fractions 3. Addition and subtraction of fractions with same denominator Example: Add

3a − 7b 2a + 2b with 15ab 15ab

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3a − 7b 2a + 2b 3a − 7b + 2a + 2b + = 15ab 15ab 15ab 3a + 2a − 7b + 2b 5a − 5b 5 × (a − b) a − b = = = = 3ab 15ab 15ab 5 × 3 × ab

6a a2 + 9 from 5a − 15 5a − 15 2 2 a +9 6a a + 9 − 6a − = 5a − 15 5a − 15 5a − 15 2 a − 6a + 9 (a − 3) 2 (a − 3) × (a − 3) a − 3 = = = = 5 × (a − 3) 5 × (a − 3) 5 × (a − 3) 5

Example: Subtract

x2 − 3 2 2x −1 + 2 − 2 2 x + 2x x + 2x x + 2x 2 x − 3 + 2 − (2 x − 1) x 2 − 3 + 2 − 2 x + 1 x 2 − 2 x x( x − 2) x − 2 = = 2 = = x + 2 x x( x + 2) x + 2 x2 + 2 x x2 + 2 x

Example: Simplify

3a 6x with 2x − a a − 2x 3a 6x 3a 6 x × (−1) 3a −6 x 3a − 6 + = + = + = 2 x − a a − 2 x 2 x − a (2 x − a) × (−1) 2 x − a a − 2 x a − 2 x Exercises

Example: Add

4. Addition and subtraction of fractions with different fractions We’ll first equalize denominators then add numerators. x 5 with Example: Add 3 4a b 6ab 4 x 5 x × 3b3 5 × 2a 2 3b3 x + 10a 2 + = + = 4a 3b 6ab 4 4a 3b × 3b3 6ab 4 × 2a 2 12a 3b 4 9

a+3 b−3 − 2 a + ab ab + b 2 a+3 b−3 (a + 3) × b (b − 3) × a − = − a ( a + b) b( a + b) a ( a + b) × b b( a + b) × a (a + 3)b − (b − 3)a ab + 3b − ba + 3a = = a(a + b)b ab(a + b) 3b + 3a 3(a + b) 3 = = = ab(a + b) ab(a + b) ab

Example: Evaluate

a2 − 3 a +1 2 2 a − 3 a − 1 a − 3 (a − 1) × (a + 1) a 2 − 3 a −1− = − = − 1 1× (a + 1) a +1 a +1 a +1

Example: Evaluate a − 1 −

a 2 − 1 a 2 − 3 a 2 − 1 − (a 2 − 3) − = a +1 a +1 a +1 2 2 a −1− a + 3 2 = = a +1 a +1 Exercises =

Product and quotient of fractions 5. Multiplication and power of fractions We’ll multiply numerators with numerators and denominators with denominators. 6b a3 Example: Multiply 2 with 2 a 4b 3 3 3 a 6b a × 6b 6a b 2 × 3 × a 2 × a × b 3a × = = = = 4b 2 a 2 4b 2 × a 2 4b 2 a 2 2 × 2 × b × b × a 2 2b Example: Multiply

pm + 2 p pm 2 with 2 m m −4 10

pm + 2 p pm 2 ( pm + 2 p ) × pm 2 × 2 = m m −4 m × (m 2 − 4) =

p × (m + 2) × p × m × m p2m = m × (m − 2) × (m + 2) (m − 2)

x −1 with x+2 x − 1 x + 1 ( x − 1) × ( x + 1) × = = x+2 x ( x + 2) × x

Example: Multiply

x +1 x x2 − 1 x2 + 2 x

x+a with x 2 − a 2 x−a x+a x + a x2 − a2 × ( x2 − a2 ) = × x−a x−a 1 2 2 ( x + a ) × ( x − a ) ( x + a )( x + a )( x − a ) = = ( x − a) ×1 ( x − a) 2 = ( x + a)( x + a) = x + 2ax + a 2

Example: Multiply

Example: Evaluate third power of (

2a 2 b4

2a 2 3 (2a 2 )3 23 × (a 2 )3 8a 6 ) = 4 3 = = 12 b4 (b ) b12 b

Exercises

6. Division of fractions We reverse second fraction then multiply them. 7a 2 14a Example: Divide 9 with b b 2 7a 14a 7a 2 b 7a 2 × b a : = × = = 8 9 9 9 b b b 14a b × 14a 2b 11

Example: Divide

Example: Divide

x−2 x +1 with x x+2 x − 2 x + 1 x − 2 x + 2 x2 − 4 : = × = x x+2 x x + 1 x2 + x a2 − 9 with a + 3 3y

a2 − 9 a2 − 9 a + 3 a2 − 9 1 a2 − 9 : (a + 3) = : = × = 3y 3y 1 3y a + 3 3 y × (a + 3) (a − 3)(a + 3) a − 3 = = 3 y (a + 3) 3y Exercises

7. Evaluation rational expressions 1 x2 − 4 × x+2 x 2 x + 1 1× ( x − 4) x + 1 ( x + 2)( x − 2) − = − 1 ( x + 2) × x 1 ( x + 2) x x + 1 x − 2 ( x + 1) × x x − 2 = − = − 1 1× x x x 2 ( x + 1) x − ( x − 2) x + x − x + 2 x 2 + 2 = = = x x x

Example: Evaluate x + 1 −

b a a 2b + ab 2 + ) × +1 a 2 − ab ab − b 2 a 2 + b2 b a a 2b + ab 2 ( + )× 2 +1 a ( a − b) b( a − b) a + b2

Example: Evaluate (

=(

b×b a×a a 2b + ab 2 + )× 2 + 1e a ( a − b) × b b( a − b) × a a + b2

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=(

b × b + a × a a 2b + ab 2 b 2 + a 2 a 2b + ab 2 )× 2 + 1 = × +1 a (a − b)b a + b2 a (a − b)b a 2 + b 2 =

(b 2 + a 2 ) × (a 2b + ab 2 ) +1 a (a − b)b × (a 2 + b 2 )

a 2b + ab 2 ab (a + b) a+b +1 = +1 = +1 a (a − b)b a ( a − b) b a−b a + b 1 a + b 1× ( a − b) a + b a − b = + = + = + a − b 1 a − b 1× ( a − b) a − b a − b a +b+ a −b 2a = = a −b a −b =

x y − y x Example: Evaluate x y + −2 y x x2 − y 2 x y x2 y 2 − − x 2 − y 2 x 2 + y 2 − 2 xy xy y x yx xy = = = 2 : 2 2 2 x y + − x y 2 xy x y 2 xy xy xy + −2 + − y x xy xy xy xy = =

x2 − y2 xy ( x 2 − y 2 ) × xy × 2 = xy x + y 2 − 2 xy xy × ( x 2 + y 2 − 2 xy ) ( x 2 − y 2 ) xy xy ( x + y − 2 xy ) 2

2

=

( x + y )( x − y ) x + y = x− y ( x − y)2

Exercises

CHAPTER II Radicals Content Real numbers Radicals

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Properties of radicals Application of radicals

Real numbers 9. Rational numbers Natural numbers are 1, 2, 3, 4, … and their symbol is Ν . Integers are ..., -3, -2, -1, 0, 1, 2, 3, … and their symbol is Z . m Rational numbers are in the form , m is integer and n is natural , n their symbol is Q. We can write rational numbers with decimal numbers. 1 Examples: is a rational number after dividing 1 with 8. We can 8 1 write = 1, 25 8 2 2 is a rational number after division we can write = 0, 4 5 5 3 3 1 is a rational after division we can write 1 = 1,15 20 20 8 8 is a rational after division we can write = 0, 216216216... 37 37 0, 216216... is a repeating decimal we can write 0, (216) or 0, 216 7 7 = 0,58333... we can write = 0,58(3) 12 12 1 5 = 5,1666... = 5,1(6) 6 5 − = −0, (45) 11 Also we can write all decimals as repeating decimals 2,5 = 2,5000... = 2,5(0) −3 = −3, 000... = −3, (0) 14

Exercises

10. Irrational numbers 3, 0100100010000... is an irrational number. −5, 020022000222... is an irrational number. π = 3,1415926535897932384626433832795... is an irrational number. e = 2,7182818284590452353602874713527... is an irrational number. 2 = 1,4142135623730950488016887242097... is an irrational number. 0,123456789101112... is an irrational number. We can not write fractions for irrational numbers. But we can write 12334 ) for rational numbers. ( 0,12334 = 100000 Rational and irrational numbers make real numbers. Exercises

Radicals 11. Square roots x 2 = 64 is an equation. Roots are 8 and −8 . So square root of 64 is 8 . With square roots we use positive root. 4 = 2 , , 0 = 0 , 1, 21 = 1,1 a = b if b ≥ 0 and b 2 = a a = undefined if a < 0

−25 = undefined ,

−3, 7 = undefined

( a ) 2 = a if a > 0 Exercises 15

12.

x 2 = a equation.

In solution of x 2 = a , there are three cases. Case1. if a < 0 , equation x 2 = a has no root. Case2. if a = 0 , equation x 2 = a has one root, this root is 0. Case3. if a > 0 , equation x 2 = a has two roots. These are x1 , x2 x1 = a and x2 = − a Example: Solve x 2 = 49 x1 = 49 = 7 and x2 = − 49 = −7 4 Example: Solve x 2 = 9 4 2 4 2 x1 = = and x2 = − =− 9 3 9 3 2 Example: Solve x = 2 x1 = 2 and x2 = − 2

Exercises

13. Calculating value of square root approximately Example: Calculate 2 12 = 1 and 22 = 4 so answer must be between 1 and 2, i.e. 1,12 = 1, 21 1, 22 = 1, 44 1,32 = 1,96 1, 42 = 1,96 1,52 = 2, 25 so answer is between 1,4 and 1,5. i.e. 1, 412 = 1,9881 1, 422 = 2, 0164 so answer is between 1,41 and 1,42. i.e. And we can continue if we want.

2 = 1,...

2 = 1, 4... 2 = 1, 41...

Exercises

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14.

y = x and its graph

We use some values of variables. x 0 1 4 9 y 0 1 2 3

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Properties of graphic. 1. x = 0 and y = 0 graphic passes through origin. 2. x ≥ 0 and y ≥ 0 graphic is on first quadrant. 3. x and y increase together, graphic increases.

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y

x -1 -1

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Properties of square root 15. Square root of product and fraction Calculate 81× 4 and 81 − 4 81× 4 = 324 = 18 81 − 4 = 9 − 2 = 7 Theorem: if a ≥ 0 and b ≥ 0 then

a×b = a × b

a a = b b 64 × 0, 04 = 64 × 0, 04 = 8 × 0, 2 = 1, 6

Theorem2: if a ≥ 0 and b > 0 then Example:

Example: 32 × 98 = 16 × 2 × 2 × 49 = 16 × 4 × 49 = 16 × 4 × 49 = 4 × 2 × 7 = 56 Example: Example:

36 36 6 = = 169 169 13 20 × 5 = 20 × 5 = 100 = 10 17

Example:

80 80 = = 16 = 4 5 5

Exercises

16. Square root of power For x = 5 and x = −6 calculate For x = 5 , For x = −6 ,

x2

x 2 = 52 = 25 = 5

x 2 = (−6) 2 = 36 = 6

Or shortly we can write

52 = 5 = 5 and

Theorem:

x2 = x

Example:

a16 = (a8 ) 2 = a8

(−6) 2 = −6 = 6

But we know even power of any number is not negative so a16 = (a8 ) 2 = a8 = a8 Example:

x10 = ( x 5 ) 2 = x5

If x ≥ 0 then x 5 is positive or zero so

x10 = ( x 5 ) 2 = x 5 = x 5

If x < 0 then x 5 is negative and − x 5 is positive so x10 = ( x5 ) 2 = x5 = − x 5 Example:

893025 = 36 × 52 × 7 2 = (33 ) 2 × 52 × 7 2

= 33 × 5 × 7 = 945 Exercises

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Application of square root 17. Moving the factor to inside or outside of root Compare 50 and 6 2 First way 50 = 25 × 2 = 5 × 2 = 5 2 , 5 2 < 6 2 so 50 < 6 2 Second way 6 2 = 6 × 2 = 36 × 2 = 36 × 2 = 72 50 < 72 so 50 < 6 2 Example:

a 7 = a 6 × a = a 6 × a = (a 3 )2 × a

= a3 × a = a3 a Example: −4 x = −4 × x = −1× 4 × x = −1× 16 × x = −1× 16 × x = −1× 16 x = − 16 x Example: a 2 If a ≥ 0 then a 2 = a 2 = a 2 × 2 = a 2 × 2 = 2a 2 If a < 0 then a 2 = − a 2 = − a 2 × 2 = − a 2 × 2 = − 2a 2 Exercises

18. Evaluation of expressions with square roots Example: 3 5a − 20a + 4 45a = ? = 3 5a − 4 × 5a + 4 9 × 5a = 3 5a − 2 5a + 12 5a = (3 − 2 + 12) 5a = 13 5a Example: (3 5 − 6 2) × ( 5 + 2 2) = ? 19

= 3 5× 5 +3 5×2 2 −6 2× 5 −6 2×2 2 2

2

= 3 5 + 6 10 − 6 10 − 12 2 = 3 × 5 − 12 × 2 = 15 − 24 = −9 Example: Simplify

x2 − 3 x+ 3

2

x2 − 3 x2 − 3 ( x + 3)( x − 3) = = = x− 3 x+ 3 x+ 3 x+ 3 c Example: Simplify 2 c c× 2 c 2 c 2 = = = 2 2 2 2× 2 2

Example: Simplify

4−3 6 6 −1

4 − 3 6 (4 − 3 6) × ( 6 + 1) 4 6 + 4 − 3 6 × 6 − 3 6 × 1 = = 2 6 −1 ( 6 − 1) × ( 6 + 1) 6 − 12 4 6 − 3× 6 − 3 6 6 − 18 = 6 −1 5 Exercises =

CHAPTER III Quadratic equations Content Quadratic equation and its roots Formula for roots of quadratic equation Rational equations

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Quadratic equation and its roots 19. Definition of quadratic equation and incomplete quadratic equation − x 2 + 6 x + 1, 4 = 0 and 8 x 2 − 7 x = 0 and x 2 −

4 = 0 are quadratic 3

equations. Quadratic equations have the form ax 2 + bx + c = 0 here x is variable, a, b, c are constants. c = 1, 4 In − x 2 + 6 x + 1, 4 = 0 a = −1 b=6 2 In 8 x − 7 x = 0 a =8 b = −7 c=0 4 4 a =1 b=0 In x 2 − = 0 c=− 3 3 These equations are quadratic equations. Definition: ax 2 + bx + c = 0 is quadratic equation if a ≠ 0 and b, c are constants and x is variable. Also we say quadratic equation as second degree equation. In incomplete quadratic equation b = 0 or c = 0 −2 x 2 + 7 = 0 and 3x 2 − 10 x = 0 and −4 x 2 = 0 are incomplete quadratic equations. Because in −2 x 2 + 7 = 0 , b=0 2 In 3x − 10 x = 0 , c=0 2 In −4 x = 0 , b = 0,c = 0 There are three types of incomplete quadratic equations. These are 1. ax 2 + c = 0 c≠0 2 2. ax + bx = 0 b≠0 2 3. ax = 0 Example: Solve −3x 2 + 15 = 0 −3x 2 + 15 = 0 −3x 2 = −15 x 2 = −15 : (−3) x2 = 5 21

x = 5 or x = − 5

Example: Solve 4 x 2 + 3 = 0 4 x2 + 3 = 0 4 x 2 = −3 3 x2 = − 4 x = noroot Example: Solve 4 x 2 + 9 x = 0 4 x2 + 9 x = 0 x × (4 x + 9) = 0 x = 0 or 4 x + 9 = 0 x = 0 or 4 x = −9 9 x = 0 or x = − 4 2 Example: Solve 5 x = 0 5x2 = 0 x2 = 0 x=0 Exercises

20. Solving quadratic equations with using square of binomials Let’s study complete quadratic equation. We’ll start with quadratic equation with a = 1 These equations are called simplified quadratic equation. Example: Solve x 2 + 10 x + 25 = 0 x 2 + 10 x + 25 = 0 ( x + 5) 2 = 0 x = −5 22

Example: Solve x 2 − 6 x − 7 = 0 x2 − 6 x − 7 = 0 x2 − 6 x = 7 x2 − 6 x + 9 = 7 + 9 ( x − 3) 2 = 16 x −3 = 4 or x − 3 = −4 x=7 or x = −1 2 Example: Solve x + 8 x − 1 = 0 x2 + 8x = 1 x 2 + 8 x + 16 = 1 + 16 ( x + 4) 2 = 17

x + 4 = 17 or x + 4 = − 17 x = 17 − 4 or x = −4 − 17 2 Example: Solve x − 4 x + 10 = 0 x 2 − 4 x = −10 x 2 − 4 x + 4 = −10 + 4 ( x − 2) 2 = −6 x = noroot Example: Solve 3x 2 − 5 x − 2 = 0 1 1 × (3x 2 − 5 x − 2) = × 0 3 3 1 1 2 × 3x 2 − × 5 x − = 0 3 3 3 5 2 x2 − x − = 0 3 3 5 2 x2 − x = 3 3 5 5 2 5 x2 − x + ( )2 = + ( )2 3 6 3 6 23

5 2 25 ( x − )2 = + 6 3 36 5 49 ( x − )2 = 6 36 5 7 5 7 x− = or x − = − 6 6 6 6 7 5 7 5 x= + or x = − + 6 6 6 6 −7 + 5 x=2 or x = 6 1 x=2 or x = − 3 Exercises

Formula of quadratic equation 21. Solving quadratic equation with formula Let ax 2 + bx + c = 0 and a ≠ 0 −b ∓ D and D = b 2 − 4ac x= 2a If D > 0 then there are two roots. If D = 0 then there is one root. If D < 0 then there is no root. Example: 12 x 2 + 7 x + 1 = 0 D = 7 2 − 4 × 12 × 1 = 1 so D > 0 1 1 −7 ∓ 1 , x1 = − x1,2 = and x2 = − 2 × 12 3 4 2 Example: x − 12 x + 36 = 0 D = 122 − 4 × 36 = 0 −(−12) ∓ 0 x1,2 = =6 2 ×1 Example: 7 x 2 − 25 x + 23 = 0 24

D = (−25) 2 − 4 × 7 × 23 = −13 so D < 0 there is no root. Exercises

22. Solving problems with the help of quadratic equation Example: Hypotenuse of a right triangle is 20cm. one side is 4cm shorter then the other side. Find length of sides. Let short side’s length be x Then long side’s length must be x + 4 With Pythagoras formula x 2 + ( x + 4) 2 = 202 x 2 + x 2 + 8 x + 16 = 400 2 x 2 + 8 x + 16 = 400 x 2 + 4 x + 8 = 200 x 2 + 4 x + 4 = 200 − 4 ( x + 2) 2 = 196 ( x + 2) 2 = 132 x + 2 = 14 or x + 2 = −14 x = 12 or x = −16 length must be positive so x = 12 short side is 12 cm and long side is 16 cm.

23. Viet theorem i. Let ax 2 + bx + c = 0 and a ≠ 0 then b c x1 + x2 = − and x1 × x2 = a a ii. if m × n = q and m + n = − p then m and n are roots of x 2 + px + q = 0 . Example: if 3x 2 − 5 x + 2 = 0 then find sum and product of roots. b −5 5 Sum of roots is − = − = a 3 3 25

Product of roots is

c 2 = a 3

Exercises

Rational equations 24. Solving rational equations x − 1 2 x 5x + = 2 3 6 x −1 2x 5x 6×( + ) = 6× ( ) 2 3 6 3 × ( x − 1) + 2 × 2 x = 5 x 3x − 3 + 4 x = 5 x 7 x − 3 = 5x 2x = 3 x = 1,5 x −3 1 x+5 + = Example: Solve x − 5 x x( x − 5) x−3 1 x+5 x( x − 5) × ( + ) = x( x − 5) × ( ) x −5 x x( x − 5) x( x − 3) + ( x − 5) ×1 = x + 5

Example: Solve

x 2 − 3x + x − 5 = x + 5 x2 − 2 x − 5 = x + 5 x 2 − 3x − 10 = 0 D = (−3) 2 − 4 × 1× (−10) = 9 + 40 = 49 −(−3) ∓ 49 3 ∓ 7 = 2 2 x1 = 5 and x2 = −2 check these answers in original equation 5 is not answer because there is x − 5 at the denominator so answer of problem is only −2 x1,2 =

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2 1 4− x − 2 = 2 x − 4 x − 2x x + 2x 2 1 4− x − = 2 2 x − 2 x( x − 2) x( x + 2) 2 1 4− x − = ( x − 2)( x + 2) x( x − 2) x( x + 2) 2 1 4− x x( x + 2)( x − 2) × ( )=( ) × x( x + 2)( x − 2) − ( x − 2)( x + 2) x( x − 2) x( x + 2)

Example:

2

2 x − ( x + 2) = (4 − x)( x − 2) 2x − x − 2 = 4x − 8 − x2 + 2x x − 2 = 6x − 8 − x2 x2 − 5x + 6 = 0 D = (−5) 2 − 4 × 6 = 25 − 24 = 1

−(−5) ∓ 1 5 ∓ 1 = 2 2 x1 = 3 and x2 = 2 check answers in original equation 2 is not answer because there is x − 2 at denominator so answer is 3. Exercises x1,2 =

25. Solving word problems with rational equations 604. Denominator of a fraction is 3 more than numerator. After adding 7 to numerator and 5 to denominator, fraction increases

1 . 2

Find this fraction 605. Numerator of a principal fraction is 5 less than denominator. After decreasing numerator with 2 and increasing denominator with 1 16, fraction decreases . Find this fraction 3 27

606. Two cars started traveling at the same time from a city to a country with 120 km distance. Car with speed 20 km/h more than the other reached country 1 hour earlier. Find speeds of cars. 607. A bike started traveling from A to B, after 1 hour 36 minutes a motorbike started traveling from A to B. They reached B together. Bike’s speed is 32 km/h less than motorbike’s speed and distance between cities is 45 km. Find bike’s speed. 608. One skater traveled 20 km distance 20 minutes shorter than the other. This skater’s speed is 2 km/h more than the other. Find each skater’s speeds. 609. Two cars started traveling at the same time from same city to another. Because of first car’s speed was 10 km/h more than the other, it reached destination 1 hour earlier. Distance is 560 find speeds of cars. 610. Because of delay with 1 hour, train increased its speed 10 km/h to travel 720 km distance just in time. Find trains normal speed. 611. Tourist traveled 6 km opposite of stream and 15 km at lake. Travel time of lake is 1 hour more than travel time of stream. If speed of stream is 2 km/h find speed of tourist at lake. 612. Speed of a boat at lake is 15 km/h, this boat traveled 35 km with stream and 25 km opposite of stream. Travel time of with stream is equal to travel time of opposite of stream. Find speed of stream. 613. Speed of a ship at lake is 20 km/h, this ship traveled 22 km with stream and 36 km opposite of stream. Total travel time is 3 hours. Find speed of stream.

28

614. One worker finishes a task 5 hours earlier than the other. Two of them together finish same task in 6 hours. Find each worker’s alone time. 615. Two workmen finish a service in 12 days. One workman need 10 days more to finish service alone than the other workman. Find each workmen’s alone time. 616. Two boys paint a wall in 6 days. One boy need 5 days more to finish painting alone than the other boy. Find each boys’ alone time. 617. Two trains from different cities with 720 km distance met at center of distance. If train with 4 km/h more speed started 1 hour later than the other. Find each train’s speed.

Extra exercises for chapter III 649. Find five consequent integer numbers such that sum of squares of first three numbers is equal to sum of square of last two numbers. 650. Find three consequent even numbers such that sum of first two numbers is equal to square of last number. 651. Area of football field is 1800 m 2 , if length is 5 m longer than width then find length and width. 652. Square of sum of two consequent numbers is 112 greater than sum of their squares. Find these numbers. 653. Perimeter of a rectangle is 28 cm and sum areas of squares constructed on one short and one long side is 116 cm 2 . Find measurements of rectangle. 657. Width is half of length of a pool. Depth is 0,5 m. Area of bottom is 1,08 m 2 less than area of laterals. Find volume of pool. 29

658. Length of a rectangle paper is 1,5 times of width. After cutting squares with 8cm from each corner of paper, we can prepare an open paper box with 6080 cm3 volume. Find measurements of paper. 681. Train delayed for 10 min at the midpoint of A and B, to reach B at estimated time it increased its speed 12 km/h. If distance between A and B is 120 km then find train’s normal speed. 682. Train delayed 1 hour 30 min after traveling one fourth of 600 km distance. To reach destination at estimated time it increased its speed 15 km/h. Find travel time of train. 683. Tourists traveled in three tunnels with 12,5 km , 18 km and 14 km. Their speed in first tunnel was 1 km/h less than in second tunnel and 1 km/h more than third tunnel. Time they spent in third tunnel 30 min more than time in second tunnel. Find three times they spend in each tunnels. 684. A car traveled from A to B with 240 km distance. Then in return it traveled half of distance with same speed, then it increased 2 its speed 10 km/h. It spent hours less in return travel than going. 5 Find going speed of car. 685. Train traveled from A to B with 400 km distance. It returned 2 of distance with same speed, then it decreased its speed 20 5 km/h .If total travel time is 11 hours then find train’s last speed. 686. Ferryboat traveled 150 km with stream and returned with opposite stream. If total travel time is 5 hours 30 minutes and speed of ferryboat at lake is 55 km/h then find stream’s speed. 687. Tourist traveled 25 km opposite of stream by motorized boat , and returned by boat. Time with motorized boat is 10 hour less than 30

time with boat. If speed of motorized boat on lake is 12 km/h find stream’s speed. 688. Ship traveled 35 km opposite of stream, and 18 km opposite of stream at a branch. It spent totally 8 hours. Stream’s speed is 1 km/h less than stream’s speed at branch. If ship’s speed at lake is 10 km/h then find stream’s speed. 689. A boat is sent with stream and after 5 hours 20 minutes a motorized boat sent from same point. Motorized boat reached boat after 20 km. If speed of motorized boat is 12 km/h more than boat then find speed of boat. 690. A man with motorized boat traveled opposite of stream 6 km then he turned off motor and stream took him to start point. If total time is 4 hours 30 min and speed of motorized boat at lake is 90 m/min then find speed of stream. 691. From port A to port B a raft is sent with stream, after 2 hours and 40 min from port B to port A a ship is sent opposite of stream. They met 27 km far from port B. If speed of ship at lake is 12 km/h and distance between A and B is 44 km then find speed of raft. 692. Truck started to go from A to B with distance 225 km. After 1,5 1 hours it delayed for hour, so it increased its speed 10 km/h to reach 2 at estimated time. Find truck’s normal speed. 693. Two cars started to travel from A to B with distance 120 km. 3 One car didn’t change its speed, other traveled hours and delayed 4 15 min and increased its speed 5 km/h. If they reached B at same time then find first car’s speed.

31

694. A bus traveled from A to B with distance 400 km. In return it traveled for 2 hours with its normal speed, then it increased its speed 10 km/h. If return time is 20 min less than going time then find return time. 695. A plane traveled from A to B in 4 hours. In return in first 100 km it traveled with normal speed, then decreased its speed 10 km/h. If return time is 30 min more than going time then find distance between A and B. 697. A plane traveled from A to B with wind in 6 hours. One time plane returned from C which distance with B 40 km and spent totally 9 hours. If speed of wind is 2 km/h find speed of plane without wind. 698. Motorbike traveled from M to N in 5 hours. In return it traveled with normal speed in first 36 km, and remaining part with 3 km/h more than normal speed. If return time is 15 minutes less than going time then find normal speed. 699. Father and son walked 240 m. Father stepped 100 less than son. Step of father is 20 cm more than step of son. Find length of father’s and son’s step. 700. First boy’s homework was 160 questions, second boy’s homework was 25% less than first boy’s homework. First boy solved 10 more questions than second boy everyday and finished 2 days earlier than estimated time. If other boy finished 2 days later than estimated time, find second boy’s daily questions. 2 of field in 4 days. If one tractor can alone 3 hoe this field 5 days earlier than the other. Find each tractors alone hoe time.

702. Two tractors hoed

707. Two workers can complete together task in 12 days. If one worker works alone at first half of task and other works alone at 32

second half then task finishes in 25 days. Find each workers alone complete time.

CHAPTER IV Inequalities Content Numerical inequalities Inequalities with one variable, system of inequalities

Numerical inequalities After comparing two numbers, we can write an equation with symbol = or an inequality with symbol < or >. For arbitrary values of variables a and b . Only one of below is true. a=b a>b a because > because 35 > 32 8 7 56 56 2. Let’s compare 3, 6748 and 3, 675 . First three digits of both numbers are same. Digits at millionths are 4 and 5 so 3, 6748 < 3, 675 9 9 and 0, 45 . After writing as decimal i.e. 20 20 9 45 9 = = 0, 45 so these are same. = 0, 45 20 100 20 4. Let’s compare −15 and −23 . Absolute value of first number is less than second number’s absolute value. So first number is greater than second number i.e. −15 > −23 3. Let’s compare

We compared some numbers in previous examples with different methods but there is a common method for comparing number too. 33

This method is calculating difference of two numbers and examining result if negative or positive or zero. Definition: if a − b is positive then a is greater than b .,if a − b is negative then a is less than b . Remark: if a − b is zero than a is equal to b . Note: if a > b then a is at right of b on number line. If a < b then a is at left of a on number line. Example1: Let’s prove with arbitrary value of a ,this inequality is true. (a − 3)(a − 5) < (a − 4) 2 we’ll write difference of two sides. (a − 3)(a − 5) − (a − 4) 2 = a 2 − 5a − 3a + 15 − (a 2 − 8a + 16) = a 2 − 8a + 15 − a 2 + 8a − 16 = −1 difference is negative so left side is less than right side. (a − 3)(a − 5) < (a − 4) 2

Example2: Let’s prove sum of squares of two arbitrary numbers is not less than twice of product of them. We want to prove a 2 + b 2 ≥ 2ab We’ll write difference of two sides a 2 + b 2 − 2ab = a 2 − 2ab + b 2 = (a − b) 2 this is positive or zero so a 2 + b 2 > 2ab or a 2 + b 2 = 2ab therefore a 2 + b 2 ≥ 2ab Exercises

Exercises of Numerical inequalities 710. If value of p − q is −5,8, 0 then compare p and q 711. Compare a and b if a) a − b = −0, 001 b) a − b = 0

c) a − b = 4,3 34

712. a < b is given. Can a − b be 3,72 or −5 or 0 713. 3a(a + 6) and (3a + 6)(a + 4) are two expressions. If a = −5 , a = 0 , a = 40 then compare values of expressions. Prove with arbitrary values of a first expression’s value is less than second expressions value. 714. 4b(b + 1) and (2b + 7)(2b − 8) are two expressions. If b = −3 , b = −2 , b = 10 then compare values of expressions. Is it possible to state with arbitrary values of a first expression’s value is more than second expressions value? 715. Prove with arbitrary value of variable inequality is true. a) 3(a + 1) + a < 4(2 + a ) b) (7 p − 1)(7 p + 1) < 49 p 2 c) (a − 2) 2 < a(a − 4) d) (2a + 3)(2a + 1) > 4a (a + 2) 716. Prove inequality a) 2b 2 − 6b + 1 > 2b(b − 3) c) p ( p + 7) > 7 p − 1

b) (c + 2)(c + 6) < (c + 3)(c + 5) d) 8 y (3 y − 10) < (5 y − 8) 2

717. Is inequality true for arbitrary values of x b) (5 x − 1)(5 x + 1) < 25 x 2 + 2 a) 4 x( x + 0, 25) > (2 x + 3)(2 x − 3) c) (3x + 8)2 > 3x( x + 16) (7 + 2 x)(7 − 2 x) < 49 − x(4 x + 1) 718. Prove inequality a) a(a + b) ≥ ab c) 2bc ≤ b 2 + c 2

d)

b) m 2 − mn + n 2 ≥ mn d) a(a − b) ≥ b(a − b)

719. Prove with arbitrary value of variable inequality is true. a) 10a 2 − 5a + 1 ≥ a 2 + a b) a 2 − a ≤ 50a 2 − 15a + 4

35

720. Prove sum of an arbitrary positive number and its reciprocal is not less than 2. 722. With showing square prove inequality b) b 2 + 70 > 16b a) a 2 − 6a + 14 > 0 723. Prove if a ≥ 0 and b ≥ 0 then

a+b ≥ ab 2

724. Which one is greater? a 2 + b 2 or ab(a + b) here a and b are different positive numbers. 725. We add same k to each 0,1,2,3. Compare product of new exteriors and product of new interiors.

28. Properties of numerical inequalities Let’s study theorems about properties of inequalities. Theorem1: if a < b then b > a and if a > b then b < a . Indeed if difference of a and b is positive then difference of b and a is negative. Theroem2: if a < b and b < c then a < c Let’s write difference of a and c. a − c = a − c + b − b = a − b + b − c = (a − b) + (b − c) is negative, because a − b is negative ( a < b ) and b − c is negative ( b < c ) and sum of two negative numbers is negative ( (a − b) + (b − c) ) So a is less than c. so a < c . After this proof it is obvious if a > b and b > c than a > c . Theorem3: if a < b then a + c < b + c , Proof; let’s write difference of a + c and b + c (a + c) − (b + c) = a + c − b − c = a − b 36

is negative because a < b given. So after adding a number to both sides of a true inequality we get another true inequality. Theorem4: if a < b and c is positive number then ac < bc And if a < b and c is negative number then ac > bc Proof: let’s write difference of ac and bc. ac − bc = c(a − b) = c × negativenumber ( a < b is given) if c is negative then c × negativenumber is positive so ac − bc is positive therefore ac > bc if c is positive then c × negativenumber is negative so ac − bc is negative therefore ac < bc This theorem is true for division of c . So we can say sign of inequality doesn’t change after multiplying or dividing both sides with same positive number and sign of inequality changes after multiplying or dividing both sides with same negative number 1 1 Corollary: if a and b are positive and a < b than > a b Proof: let’s start with a < b inequality and divide both side with ab (ab is positive) a a b Example: if 54, 2 < a < 54,3 is given and a is length of one side of equilateral triangle. Find this triangle’s perimeter. Solution: We can perimeter with formula P = 3a 37

We must multiply both sides of inequalities with 3. 54, 2 < a 3 × 54, 2 < 3 × a 162, 6 < 3a and a < 54,3 3 × a < 3 × 54,3 3a < 162,9 so 162, 6 < 3a < 162,9

Exercises of Properties of numerical inequalities 729. Draw number line. Mark points a,b,c,d and e if a < b and c > b and c < d and a > e 730. m,n,p and q are numbers. If m > p and n > m and n < q then compare p and n, p and q, q and m if it is possible. Use number line. 731. a < b is given. If it is possible compare a and b+1, a − 3 and b, a − 5 and b+2, a+4 and b − 1 . 732. If inequalities below are true determine sign of a and b (positive or negative) a) a − 3 > b − 3 and b > 4 b) a − 8 > b − 8 and a < −12 1 c) 7 a > 7b and b > 1 d) −2a > −2b and b < − 3 733. With using properties of inequalities evaluate a) Add −5and 2, 7 and 7 to both sides of inequality 18 > −7 b) Subtract 2and12and − 5 from boths sides of inequality 5 > −3 1 c) Multiply 2and − 1and − to both sides of inequality −9 < 21 3 d) Divide both sides of inequality 15 > −6 with 3and − 3and − 1 38

734. a < b is given. With using properties of inequality a) Add 4 to both sides b) Subtract 5 from both sides c) Multiply both sides with 8 1 d) Divide both sides with 3 e) Multiply both sides with (−4,8) f) Divide both sides with −1 735. a < b is given. Change ? sign with < or > to make true inequality. a b a) −12, 7 a ?− 12, 7b b) ? 3 3 a b c) 0, 07a ? 0, 07b d) − ?− 2 2 736. Determine the sign of a in inequalities below. b) 7 a > 3a c) −3a < 3a d) −12a > −2a a) 5a < 2a 738. a, b, c, d are positive numbers and a > b and d < b and c > a , 1 1 1 1 Order , , , in increasing order. a b c d 739. 3 < a < 4 is given. Determine value of expression. b) − a c) a + 2 d) 5 − a a) 5a 0, 2a + 3

e)

740. 6 < x < 8 is given. Determine value of expression. b) −10x c) x − 5 d) 3 x + 2 a) 6x 741. Use 1, 4 < 2 < 1,5 determine value of expression. a)

2 +1

b)

2 −1

c) 2 − 2

39

742. Use 2, 2 < 5 < 2,3 and determine value of expression a)

5+2

b) 3 − 5

744. Determine value of a) 5 < y < 8

1 y b) 0,125 < y < 0, 25

29. Addition and multiplication of numerical inequalities Let’s study theorems about addition and multiplication of numerical inequalities. Theorem5: if a < b and c < d then a + c < b + d Proof: let’s add c to both sides of a < b a
Therefore after multiplying corresponding sides of inequalities of positive numbers with same symbol we get another true inequality Note: if there are negative numbers in a, b, c, d with a < b and c < d then inequality ac < bd may not be true. Example: let’s take a = −3 , b = −2 , c = −5 , d = 6 a < b is true because −3 < −2 c < d is true because −5 < 6 but ac < bd is not true because (−3) × (−5) < (−2) × 6 ; 15 < −12 is not true. Corollary: If a and b are positive number and a < b then a n < b n (n is natural number) Proof: Calculate product of n times a < b inequality. We use these theorems in calculating sum, difference, product and quotient of variables with inequalities. Example: 15 < x < 16 , 2 < y < 3 are given. Find x + y , x − y , xy ,

x y

1. for x + y we’ll add two inequalities. 15 < x < 16 2< y<3 17 < x + y < 29 2. for x − y firstly we’ll write an inequality with − y then we’ll add inequalities with x and − y ,so we must multiply 2 < y < 3 with (−1) 2< y<3 (−1) × 2 > (−1) × y > (−1) × 3 −2 > − y > −3 −3 < − y < −2 15 < x < 16 −3 < − y < −2 15 − 3 < x − y < 16 − 2 3. for xy here x and y are positive numbers so xy is also positive. So we can multiply both inequalities. 41

15 < x < 16 2< y<3 15 × 2 < x × y < 16 × 3 30 < xy < 48 x 1 4. we’ll use product of x and , so firstly we must write an y y 1 inequality with y 2< y<3 1 1 1 > > 2 y 3 1 1 1 < < 3 y 2 15 < x < 16 1 1 1 < < 3 y 2 1 1 1 15 × < x × < 16 × 3 y 2 x 5< <8 y

Exercises of Addition and multiplication of numerical inequalities 747. Add inequalities term by term b) −2,5 < −0, 7 and −6,5 < −1,3 a) 12 > −5 and 9 > 7 748. Multiply inequalities term by term a) 5 > 2 and 4 > 3

b) 8 < 10 and

1 1 < 4 2

749. Is it true for positive numbers a and b. a) if a > b then a 2 > b 2 42

b) if a 2 > b 2 then a > b 750. Find value of expressions if 3 < a < 4 and 4 < b < 5 are given. a d) b) a − b c) ab a) a + b b 751. Find value of expressions if 6 < x < 7 and 10 < y < 12 are given. y a) x + y b) y − x c) xy d) x 752. Find value of expressions if 1, 4 < 2 < 1,5 and 1, 7 < 3 < 1,8 a)

2+ 3

b)

3− 2

753.Find value of expression if 2, 2 < 5 < 2,3 and 2, 4 < 6 < 2,5 a)

6+ 5

b)

6− 5

754. Two sides of an isosceles triangle are a and b. If 26 ≤ a ≤ 28 and 41 ≤ b ≤ 43 are given then find perimeter of triangle. 755. Two sides of a right triangle are a and b. If 5, 4 < a < 5,5 and 3, 6 < b < 3, 7 are given then find a) area of triangle b) perimeter of triangle 756. Width and length of a room are a and b. Is this room suitable for a bookstore with area no less than 40 m2 . If 7,5 ≤ a ≤ 7, 6 and 5, 4 ≤ b ≤ 5,5 757. α and β are two angles of a triangle , and 58 ≤ α ≤ 59 , 103 ≤ β ≤ 120 . Find value of third angle.

43

Inequalities with one variable and system of inequalities 30. Intervals inequality −3 < x < 2

−3 ≤ x ≤ 2 −3 ≤ x < 2 −3 < x ≤ 2

number line -3 2 -3

2

-3

2

-3

2

x>6 x≥6

6

x ≤ 10 x: all real numbers

(−3, 2) [−3, 2] [−3, 2) (−3, 2]

(6, +∞)

6

x < 10

interval

[6, +∞) 10 10

(−∞,10)

(−∞,10] (−∞, +∞)

Intersection of intervals [1,5] ∩ [3, 7] = [3,5] Union of intervals [1,5] ∪ [3, 7] = [1, 7]

Exercises of Intervals 761. Show intervals below on number line a) [−2; 4] c) [0;5] e) (3; +∞)

g) (−∞; 4] 44

b) (−3;3)

d) (−4;0)

f) [2; +∞)

h) (−∞; −1)

762. Show interval on number line b) [1;6] c) (−∞;5) a) (3;7)

d) [12; +∞)

763. Show numbers on number line that satisfies inequality a) x ≥ −2 b) x ≤ 3 c) x > 8 d) x < −5 764. Show numbers on number line that satisfies inequality 1 a) −1,5 ≤ x ≤ 4 c) −5 ≤ x ≤ −3 3 b) −2 < x < 1,3 d) 2 < x ≤ 6,1 765.

a) Which of −3; −5;5; −6; −7,5 belongs (−4;6,5) b) Which of −9; −8; −5,5; −5; −6; −7,5 belongs [−8; −5]

766. Which of −1, 6; −1,5; −1;0;3;5,1;6,5 belongs c) (−∞; −1) a) [−1,5;6,5] b) (3; +∞) 767. Which of

2; 3; 5; 6 belongs (1,5; 2, 4)

768. Show two positive and two negative numbers in interval below b) [−1;1] a) (−4;5) 769. Write integers in interval below a) (−4;3) b) [−3;5] 770. Write integers in interval below a) [0;8] b) (−3;3) c) (−5; 2)

d) (−4;9]

771. Write the greatest integer in interval below b) [−1;17) c) (−∞;31] a) [−12; −9]

d) (−∞;8) 45

772. Is 1,98 in interval (−∞; 2) ? Write two numbers in interval that is greater than 1,98. Is it possible to find the greatest number in this interval? Is it possible to find the least number in this interval? 773. Find intersection of intervals using number line. a) (1;8) and (5;10) c) (5; +∞) and (7; +∞) b) [−4; 4] and [−6;6] d) (−∞;10) and (−∞;6) 774. Show union of intervals on number line. a) [7;0) and [−3;5] c) (−∞; 4) and (10; +∞) b) (−4;1) and (10;12) d) [3; +∞) and (8; +∞) 775. Find union and intersection of intervals a) (−3; +∞) and (4; +∞) b) (−4;1) and (10;12)

c) (−∞; 4) and (10; +∞) d) [3; +∞) and (8; +∞)

31. Solving inequalities with one variable Solving inequality is similar with solving equations. The only difference is changing of inequality sign after multiplying or dividing both sides with negative number. And also we use intervals for answers. Example: Let’s solve 16 x > 13 x + 45 16 x − 13 x > 45 3 x > 45 x > 45 : 3 x > 15 answer is (15, +∞) Example: Let’s solve 15 x − 23( x + 1) > 2 x + 11 15 x − 23 x − 23 > 2 x + 11 −8 x − 23 > 2 x + 11 −8 x − 2 x > 11 + 23 −10 x > 34 46

x < 34 : (−10) x < −3, 4 answer is (−∞, −3.4) Example: let’s solve

x x − <2 3 2 x x 6× ( − ) < 6× 2 3 2 2 x − 3 x < 12 − x < 12 x > −12 answer is (−12, +∞)

Example: Let’s solve 0 × x > 1 0 × x is always 0. 0 is not greater than 5. so no answer. Example: Solve 0 × x < 2 0 × x is always 0. 0 is less than 2. so answer all real numbers. (−∞, ∞) Example: Solve 0 × x > −3 0 × x is always 0. 0 is greater than −3 . so answer all real numbers. (−∞, ∞) Example: Solve 0 × x < −4 0 × x is always 0. 0 is not less than −4 . so no answer.

Exercises of Solving inequalities with one variable 780. Which of 8; −2;1,5; 2 is root of 5 y > 2( y − 1) + 6 781. Which of −2; −1; −1,5; −0,3 is root of 12 x + 4 < 7 x − 1 782. Show two roots of 2 x < x + 7 783. Solve inequality and show on number line 47

a) x + 8 > 0

b) x − 7 < 0

c) x + 1,5 ≤ 0

784. Solve inequality a) 3 x > 15 b) −4 x < −16

d) 12 y < 1,8 e) 27b ≥ 12

c) − x ≥ −1

f) −6 x > 1,5

d) 11 y ≤ 33

g) 15 x ≤ 0

d) x − 04 ≥ 0

h) 0,5 y > −4 i) 2,5a > 0 1 j) x > 6 3 1 k) − y < −1 7

785. Solve inequality and show on number line a) 2 x < 17

e) 30 x > 40

b) 5 x ≥ −3

f) −15 x < −27

c) −12 x < −48 d) − x < −7,5

g) −4 x ≥ −1 h) 10 x ≤ −24

1 x<2 6 1 j) − x < 0 3 k) 0, 02 x ≥ −0, 6 l) −1,8 x ≤ 36 i)

786. Solve inequality 5 x + 1 > 11 , write three roots 4 4 787. Solve inequality 3 x − 2 < 6 . Are 4; 2 ; 2 roots ? 5 7 788. Solve inequality a) 7 x − 2, 4 < 0, 4 b) 1 − 5 y > 3 c) 2 x − 17 ≥ −27 d) 2 − 3a ≤ 1

e) 17 − x > 10 − 6 x f) 30 + 5 x ≤ 18 − 7 x g) 64 − 6 y ≥ 1 − y h) 8 + 5 y ≤ 21 + 6 y

48

789. Solve inequality and show on number line e) 3 y − 1 > −1 + 6 y a) 11x − 2 < 9 b) 2 − 3 y > −4 f) 0, 2 x − 2 < 7 − 0,8 x c) 17 − x ≤ 11 g) 6b − 1 < 12 + 7b h) 16 x − 34 > x + 1 d) 2 − 12 > −1 790. a) With which values of x expression 2 x − 1 takes positive values. b) With which values of y expression 21 − 3y takes negative values. c) With which values of c expression 5 − 3c takes values greater than 80. 791. a) With which values of a value of 2a − 1 is less than value of 7 − 1, 2a . b) With which values of p value of 1,5 p − 1 is greater than value of 1 + 1,1p 792. Solve inequality a) 5( x − 1) + 7 ≤ 1 − 3( x + 2) b) 4(a + 8) − 7(a − 1) < 12 c) 4(b − 1,5) − 1, 2 ≥ 6b − 1 d) 1, 7 − 3(1 − m) ≤ −(m − 1,9)

e) 4 x > 12(3x − 1) − 16( x + 1) f) a + 2 < 5(2a + 8) + 13(4 − a) g) 6 y − ( y + 8) − 3(2 − y ) ≤ 2

793. Solve inequality a) 4(2 − 3x) − (5 − x) > 11 − x b) 2(3 − z ) − 3(2 + z ) ≤ z c) 1 > 1,5(4 − 2a) + 0,5(2 − 6a) . d) 2,5(2 − y ) − 1,5( y − 4) ≤ 3 − y e) x − 2 ≥ 4, 7( x − 2) − 2, 7( x − 1) f) 3, 2(a − 6) − 1, 2a ≤ 3(a − 8) 794. Solve inequality and show on number line 49

a) a (a − 4) − a 2 > 12 − 6a b) (2 x − 1)2 x − 5 x < 4 x 2 − x

c) 5 y 2 − 5 y ( y + 4) ≥ 100 d) 6a(a − 1) − 2a(3a − 2) < 6

795. Solve inequality a) 0, 2 x 2 − 0, 2( x − 6)( x + 6) > 3, 6 x b) (2 x − 5) 2 − 0,5 x < (2 x − 1)(2 x + 1) − 15 c) (12 x − 1)(3x + 1) < 1 + (6 x + 1) 2 d) (4 y − 1) 2 > (2 y + 3)(8 y − 1) 796. Solve inequality a) 4b(1 − 3b) − (b − 12b 2 ) < 43

c) 2 p(5 p + 2) − p(10 p + 3) ≤ 14

b) 3 y 2 − 2t − 3 y ( y − 6) ≥ −2

d) a(a − 1) − (a 2 + a) < 34

797. Solve inequality 2x 3x − 1 >1 d) a) >2 5 4 6− x x b) < 2 e) 2 > 5 3 6x 2 + 3x ≥0 f) c) <0 7 18

12 − 7 x ≥0 42 1 h) ( x + 15) > 4 3 2 i) 6 ≤ ( x + 4) 7

798. Solve inequality 9x 5 + 6x ≥0 c) a) >3 5 2 3x 4 x − 11 b) 1 < d) ≤0 4 4

1 x≥2 7 2 f) ( x − 4) < 3 11

g)

e)

799. Which values of y makes 7 − 2y 3y − 7 greater than value of a) Value of 6 12 50

4,5 − 2 y 2 − 3y less than value of 5 10 3 y −1 c) Value of 5 y − 1 greater than value of 4 5− 2y less than value of 1 − 6 y d) Value of 12 b) Value of

800. Solve inequality x x a) + < 5 2 3 3y y − ≥2 b) 2 3

x x c) − > −3 4 2 y d) y + > 3 2

2x − x ≤1 5 3x f) − 2x < 0 4 e)

801. Solve inequality and show on number line x x 13x − 1 < 4x c) − ≤ 2 a) 2 4 5 5 − 2a 2y y ≥ 2a d) b) − ≥1 4 5 2 802. Solve inequality 3+ x 2− x + <0 a) 4 3 4− y − 5y ≥ 0 b) 5 2 y −1 ≥1 c) y − 4 803. Solve inequality 2a − 1 3a − 3 a) − >a 2 5 2x + 3 x −1 ≤ b) x − 2 4

x − 3 2x −1 + ≤4 5 10 y −1 2 y −1 e) −1+ ≥y 2 6 p −1 p + 3 f) p − − >2 2 4

d) x −

5x − 1 x + 1 + ≤x 5 2 y −1 2 y + 3 d) − −y>2 2 8

c)

51

804. Which value of variable makes 2a − 1 a −1 a) Sum of and positive 4 3 3b − 1 1 + 5b and negative b) Difference of 2 4 805. Solve inequality a) 21(2 x + 1) − 12 x > 50 x x 2x b) x + 4 − < 3 3

c) 3 x + 7 > 5( x + 2) − (2 x + 1) 12 − x d) < 4x − 3 3

806. Function y = −1,5 x + 7,5 is given. Which value of x makes b) y > 0 c) y < 0 a) y=0 807. Which values of x makes value of y = 2 x + 13 negative, positive. 808. With which value of variable expression is no undefined 1 + 3a e) −3(1 − 5 x) c) a) 2 x − 4 25 b)

4 − 6a

d)

7 − 5a 8

f)

−(6 − x)

809. a) Find greatest integer satisfies inequality 1, 6 − (3 − 2 y ) < 5 b) Find least integer satisfies inequality 8(6 − y ) < 24, 2 − 7 y 810. Which natural n makes a) Difference (2 − 2n) − (5n − 27) positive b) Sum (−27,1 + 3n) + (7,1 + 5n) negative

32. Solving system of inequalities with one variable We solve inequalities separately and find intervals. Then we find intersection of these intervals. 52

2 x − 1 > 6 Example1: Solve  5 − 3x > −13 2x −1 > 6 2x > 7 x > 3,5 (3.5, +∞) and 5 − 3 x > −13 −3 x > −18 x<6 (−∞, 6) The intersection of these intervals is (3.5, +∞) ∩ (−∞, 6) = (3.5, 6) or 3,5 < x < 6 3x − 2 > 25 Example2: Solve  1 − x < 0 3 x − 2 > 25 3 x > 27 x>9 (9, +∞) and 1 − x < 0 − x < −1 x >1 (1, +∞) The intersection of these intervals is (9, +∞) ∩ (1, +∞) = (9, +∞) or x>9 2 − x > 0 Example3: Solve  0, 2 x − 1 < 0 2− x > 0 − x > −2 x<2 (−∞, 2) 53

and 0, 2 x − 1 < 0 0, 2 x < 1 x < 1: 0, 2 x<5 (−∞,5) The intersection of these intervals is (−∞, 2) ∩ (−∞,5) = (−∞, 2) or x<2 1 − 5 x > 11 Example4: Solve  6 x − 18 > 18 −5 x > 10 x < 10 : (−5) x < −2 (−∞, −2) and 6 x − 18 > 18 6 x > 36 x>6 (6, +∞) The intersection of these intervals is (−∞, −2) ∩ (6, +∞) = empty so there is no answer Example5: Solve −1 < 3 + 2 x < 3  −1 < 3 + 2 x This is same with  3 + 2 x < 4 −1 < 3 + 2 x −4 < 2 x −2 > x (−2, +∞) and 3 + 2 x < 4 2x < 1 x < 0,5 (−∞, 0.5) 54

The intersection of these intervals is (−2, +∞) ∩ (−∞, 0.5) = (−2, 0.5) −2 < x < 0,5 or −1 < 3 + 2 x < 3 −1 − 3 < 2 x < 4 − 3 −4 < 2 x < 1 −2 < x < 0,5

Exercises of Solving system of inequalities with one variable 818. Is 3 root of system of inequality? 6 x − 1 < x 7 x < 5 x + 7 b)  a)  4 x − 32 < 3x 3 x − 1 > 5 − x

5 x + 4 < 20 c)  3 − 2 x > −1 3x − 22 < 0 819. Which of −2;0;5;6 is root of inequality  2 x − 1 > 3 820. Solve system of inequality. x > 0  x > 17 c)  a)   x > 12 x < 6 x < 1 b)  x < 5

 x < −3,5 d)  x > 8

821. Solve system of inequality 2 x − 12 > 0 a)  3x < 9 4 y < −4 b)  5 − y > 0

 x ≥ −1 e)  x ≤ 3 x > 8 f)   x ≤ 20

3x − 10 < 0 c)  2 x > 0 6 y ≥ 42 d)  4 y + 12 ≤ 0

822. Solve system of inequality 55

 x − 0,8 > 0 a)  5 x < 10 2 − x ≤ 0 b)  x − 4 ≤ 0

1 > 3x c)  5 x − 1 > 0 10 x < 2 d)   x > 0,1

823. Solve system of inequality 0, 6 x + 7, 2 > 0 a)  5, 2 ≥ 2, 6 x 1,5 x + 4,5 ≤ 0  b)  1  9 x ≥ 0 825. Solve system of inequality 2 x − 1 < 1, 4 − x a)  3x − 2 > x − 4

0, 2 x < 3  c)  1  6 x > 0 2 x − 6,5 < 0  d)  1  3 x < −1

5 x + 6 ≤ x b)  3x + 12 ≤ x + 17

17 x − 2 < 12 x − 1 c)  3 − 9 x < 1 − x 25 − 6 x ≤ 4 + x d)  3x + 7, 7 > 1 + 4 x

826. Solve system of inequality 57 − 7 x > 3 x − 2 a)  22 x − 1 > 2 x + 47

102 − 73z > 2 z + 2 c)  81 + 11z ≥ 1 + z

1 − 12 y < 3 y + 1 b)  2 − 6 y > 4 + 4 y

6 + 6, 2 x ≥ 12 − 1,8 x d)  2 − x ≥ 3,5 − 2 x

827. Show possible values of variable a) 3 − 2 x + 1 − x c) b)

x − 3x − 1

d)

6 − x − 3x − 9 2x + 2 + 6 − 4x

828. Solve system of inequality 56

5( x − 2) − x > 2 a)  1 − 3( x − 1) < −2  2 y − ( y − 4) < 6 b)   y > 3(2 y − 1) + 18

7 x + 3 ≥ 5( x − 4) + 1 c)  4 x + 1 ≤ 43 − 3(7 + x) 3(2 − 3 p ) − 2(3 − 2 p ) > p d)  2 6 < p − p ( p − 8)

829. Solve system of inequality 2( x − 1) − 3( x − 2) < x a)  6 x − 3 < 17 − ( x − 5) 3,3 − 3(1, 2 − 5 x) > 0, 6(10 x + 1) b)  1, 6 − 4,5(4 x − 1) < 2 x + 26,1 5,8(1 − a) − 1,8(6 − a ) < 5 c)  8 − 4(2 − 5a) > −(5a + 6)  x( x − 1) − ( x 2 − 10) < 1 − 6 x d)  3,5 − ( x − 15) < 6 − 4 x 830. Solve system of inequality and show all integer roots 3 − 2a < 13 2 − 6 y < 14 c)  a)  5a < 17 1 < 21 − 5 y

12 − 6 x ≤ 0 b)  3 x + 1 ≤ 25 − x

3 − 4 x < 15 d)  1 − 2 x > 0

831. Find integer roots of system of inequality 6 − 4b > 0 y ≥ 0 c)  a)  7, 2 − y ≥ 0 3b − 1 > 0 12a − 37 > 0 b)  6a ≤ 42

3 − 18 x < 0 d)  0, 2 − 0,1x > 0

832. Solve system of inequality 2,5a − 0,5(8 − a) < a + 1, 6 a)  1,5(2a − 1) − 2a < a + 2,9 57

0, 7(5a + 1) − 0,5(1 + a) < 3a b)  2a − (a − 1, 7) > 6, 7 833. Solve system of inequality x x  2 + 4 < 7 a)  1 − x > 0  6 y −1   y − 2 > 1 b)  y <5  3

 3x − 1  2 − x ≤ 2 c)  2 x − x ≥ 1  3 p−2  2 p − 5 > 4 d)  p− p ≤6  2 8

834. Solve system of inequality  x −1 x − 3  2 − 3 < 2 a)  13x − 1 > 0  2  3x + 1  2 < −1 b)   x −1 < x  2

y−2  4 − 3 ≥ y c)   7 y −1 ≥ 6  8  5a + 8  3 − a ≥ 2a d)  1 − 6 − 15a ≤ a  4

835. Solve inequality a) −3 < 2 x − 1 < 3 b) −12 < 5 − x < 17

c) 2 < 6 − 2 y < 5 d) −1 < 5 y + 4 < 19

836. Solve inequality 7x + 6 ≤ 20,5 a) −6,5 ≤ 2

c) −2 ≤

3x − 1 ≤0 8 58

b) −1 ≤

4−a ≤5 3

837. Solve inequality a) −1 ≤ 15 x + 14 ≤ 44 6−a ≤1 b) −1 ≤ 3

d) −2,5 ≤

1− 3y ≤ 1,5 2

c) −1, 2 ≤ 1 − 2 y ≤ 2, 4 4x −1 d) −2 < ≤0 3

838. Which values of variable makes a) Value of 3 y − 5 in (−1;1) 5 − 2b in [−2;1] b) Value of 4 839. Solve system of inequality x > 8  y < −1   b)  y < −5 a)  x > 7  x > −4 y < 4  

m > 9  c) m > 10  m < 12

840. Solve system of inequality x − 4 < 8  a) 2 x + 5 < 13 3 − x > 1 

2 x − 1 < x + 3  b) 5 x − 1 > 6 − 2 x x − 5 < 0 

841. Solve system of inequality 3 − 2a < 13  a) a − 1 > 0 5a − 35 < 0 

6 − 4 a < 2  b) 6 − a > 2 3a − 1 < 8 

q < 6  d) q < 5 q < 1 

Extra exercises from chapter IV 846. Compare m and n if difference of m − n is b) (−3,1)36 a) (−2, 7)15 59

847. Prove inequality a) (6 y − 1)( y + 2) < (3 y + 4)(2 y + 1) b) (3 y − 1)(2 y + 1) > (2 y − 1)(2 + 3 y ) 848. Is inequality true for arbitrary values of a e) (5 − a) 2 ≥ 0 a) (a − 8) 2 > 0 c) − a 2 − 2 < 0 b) a 2 + 1 > 0

f) −(a − 3) 2 ≤ 0

d) −a 2 < 0

849. Prove inequality a) ( x + 1) 2 > 4 x b) (3b + 1) 2 > 6b 850. Prove inequality a) a 2 + b 2 + 2 > 2(a + b)

c) 4( x + 2) < ( x + 3) 2 − 2 x

b) a 2 + b 2 + c 2 + 3 ≥ 2(a + b + c)

858. a,b,c,d are arbitrary numbers. If a > b , c < b and c > d then compare a and c. Compare b and d. 859. Order a + 5 , a − 7 and a + 1 in increasing order 860. Prove if a > b then a) a + 5 > b + 3

b) 1 − a < 2 − b

861. Prove if a > b > 0 then a) 5a > 4b b) 17 a > 12b

c) −4a < −2b d) −5a < −1, 2b

862. Prove a) if a ≤ b and c positive then a + c ≤ b + c b) if a ≤ b and c positive then ac ≤ bc c) if a ≤ b and c negative then ac ≥ bc 863. If a > b which of a − 1 > b − 1 , 1 − a > 1 − b , 5 − a < 5 − b is true? 60

864. 12 ≤ y ≤ 16 is given. Find value of a) −0,5y

b) 42 − 2 y

c)

1 +2 y

865. Find value of expression a) a + 2b if 0 < a < 1 and −3 < b < −2 1 b) a − b if 7 < a < 10 and 14 < b < 15 2 867. Prove a) If a ≤ b and c ≤ d then a + c ≤ b + d b) If 0 ≤ a ≤ b and 0 ≤ c ≤ d then ac ≤ bd 869. Write all integers in interval a) [−4; 4] b) (−2,5;7)

c) (4, 2;8, 2)

d) (−4;3)

870. Is there any integer in interval a) [−1,8; −1, 6] b) [−3, 7; −2, 7] 871. Write any number in interval a) (2, 4; 2,8) b) (−3,8; −3,1) c) (3,5;3, 6)

d) (−0, 2; −0,1)

872. Is 40,9 in interval [8; 41) ? Write a number in interval that is greater than 40,9. Is it possible to find the greatest number in this interval? Is it possible to find the least number in this interval? 873. Is 7,01 in interval (7;17] ? Write a number in interval that is less than 7,01. Is it possible to find the greatest number in this interval? Is it possible to find the least number in this interval? 874. Show least and greatest number in interval a) [12;37] b) [8;13) c) (11;14)

d) [3;19)

875. Is it true? a) (−5;5) ∩ (−3; 2) = (−3; 2) 61

b) (4;11) ∪ (0;6) = (4;6) c) (−∞; 4) ∪ (1; +∞) = (−∞; +∞) d) (−∞; 2) ∩ (−2; +∞) = (−2; 2) 876. a) Find intersection and union of integers and positive numbers. b) Find intersection and union of rational and irrational numbers 877. Is 4,99 root of x < 5 ? Find a number greater than 4,99 that satisfies inequality 878. Is 3,01 root of x > 3 ? Find a number less than 3,01 that satisfies inequality 879. Solve inequality a) 0, 01(1 − 3 x) > 0, 02 x + 3, 01 b) 12(1 − 12 x) + 100 x > 36 − 49 x c) (0, 6 y − 1) − 0, 2(3 y + 1) < 5 y − 4 2 1 d) (6 x + 4) − (12 x − 5) ≤ 4 − 6 x 3 6 e) (3a + 1)(a − 1) − 3a 2 > 6a + 7 f) 15 x 2 − (5 x − 2)(3x + 1) < 7 x − 8 880. Which value of a makes inequality true? a −1 a +1 1 − 2a 1 − 5a −2< −1 > +8 c) a) 4 2 4 8 3a − 1 a − 1 5a 3a − 1 2a − 1 b) − >0 d) − + <1 2 4 6 3 2 881. Solve inequality x − 0,5 x − 0, 25 x − 0,125 + + <0 a) 4 4 8

b)

5 − x 1− x − >1 3 2

882. Find all natural numbers satisfy inequality 62

a) 3(5 − 4 x) + 2(14 + x) > 0

b) ( x + 1)( x − 1) − ( x 2 − 3x) ≤ 14

883. Find x which makes 3x − 8 x −1 greater than value of a) Value of 12 4 x +1 2x + 3 less than value of b) Value of 3 6 884. Solve inequality a) 2(4 y − 1) − 5 y < 3 y + 5

b) 6(1 − y ) − 8(3 y + 1) + 30 y > −5

x > 3 892. With which value of a system of inequality has no root  x < a 893. Solve system of inequality 4 x > 1 x < 0   a) 5 x > 0 b) − x > −1 x > 9 4 x < 8  

− x < 3  c) 2 x > 10  x < −10 

3 x > −9  d)  x < −2   −2 x > 10

894. Prove there is no root in system of inequality  x2 + 1 < 0 6 x < 0 c)  a)  3 x > 0 3 x − 1 > 0 2 x − 4 > 2 x − 1 b)  5 x > 0 895. Solve system of inequality 0,3x − 1 < x + 0, 4 a)  2 − 3 x < 5 x + 1 2,5 x − 0,12 > 0, 6 x + 0, 07 b)  1 − 2 x > − x − 4

3x + 5 > 0 d)  3x + 5 < 0 3( x − 2)( x + 2) − 3 x 2 < x d)  5 x − 4 > 4 − 5 x ( x − 4)(5 x − 1) − 5 x 2 > x + 1 e)  3 x − 0, 4 < 2 x − 0, 6

63

3x − 7  2 x + 1, 4 < 5 c)  2 x > 3 − 2 x  5

 1 + x 2x −1 1 + 3 > 6 − 2 f)  3x − x > 4  4

896. Find integer roots of system of inequality 6 x( x − 1) − 3x(2 x − 1) < x a)  0,5 x − 3, 7 < 0, 2 x − 0, 7 0, 7 x − 3(0, 2 x + 1) ≤ 0,5 x + 1 b)  0,3(1 − x) + 0,8 x ≥ x + 5,3 1 1  3 (3 x − 2) + 6 (12 x + 1) > 0 c)   1 (14 x − 21) + 2 (9 x − 6) < 0  7 9 1  0, 2(5 x − 1) + 3 (3 x + 1) < x + 5,8 d)  8 x − 7 − 1 (6 x − 2) > x  6 897. Solve inequality a) −9 < 3 x < 18 2x +1 <2 b) 1 < 2

c) 3 ≤ 5 x − 1 ≤ 4 1− x d) 0 ≤ ≤1 3

898. a) Which value of x makes value of 2 x − 4 in (−1;5) x −5 in [0;5] b) Which value of x makes value of 3 1 c) Which value of x makes value of y = − x + 8 in (−1;1) 3 d) Which value of x makes value of y = −2,5 x + 6 in [−2; −6] 64

899. Find positive values of y satisfies inequality 3( y − 1) − 4( y + 8) < 5( y + 5) a)  1, 2(1 + 5 y ) − 0, 2 < 5(1 − 3 y ) − 3 y 15( y − 4) − 14( y − 3) < y ( y − 9) − y 2  b)  5 − y 2− y − y > 14 −  6  3 (2 y − 1)(3 y + 2) − 6 y ( y − 4) < 48  c)  y − 1 6 y + 1  8 − 4 − 1 < 0 900. Find negative values of y satisfies inequality 5 y −1 2 y −1  6 − 2 > 0 ( y + 6)(5 − y ) + y ( y − 1) > 0 b)  a)  2 0,3 y (10 y + 20) − 3 y + 30 > 0 1 − y + 4 < 0  3

CHAPTER V Powers with integer exponent Content Powers with integer exponent and properties Error calculations

Power with integer exponent and properties 33. Definition of power with negative integer Definition: if a ≠ 0 and n is negative integer then 1 an = −n a 1 1 1 Example: 5−2 = − ( −2) = 2 = 5 5 25

65

1 1 1 = = − ( −4) 4 (−3) (−3) 81 1 1 1 1 8 (− ) −3 = = = = − = −8 1 1 1 2 1 (− ) − ( −3) (− )3 − 2 2 8 n note : 0 = 0 if n ≥ 1 and n is integer (−3) −4 =

Exercises

Exercises of Definition of power with negative integer 903. Change negative power with fraction c) a −1 a) 10−6

e) (ab) −3

b) 9−2

f) (a + b) −4

d) x −20

904. Change fraction with negative power 1 1 1 1 d) 10 b) 7 c) 7 a) 2 y x 10 6

e)

1 7

1 1 1 905. a) Show 8; 4; 2;1; ; ; as power of 2 2 4 8 1 1 1 ; ; ;1;5; 25;125 as power of 5 b) Show 125 25 5 1 1 1 ; ; ;1;3;9; 27;81 as power of 3 81 9 3 b) Show 100;10;1;0,1;0, 01;0, 001;0, 0001 as power 10 906. a) Show

907. Calculate a) 4

−2

d) (−1) −20

 2 f)  −   3

−3

 2 h)  −2   5

−2

66

b) (−3)

−3

1 e)   7

−2

 1 g) 1   2

−5

c) (−1) −9

i) 0, 01−2 j) 1,125−1

908. Find value of expression c) (−0,8) −2 a) −10−4 b) −0, 2−3 d) (−0,5) −5

e) −(−2) −3 f) −(−3) −2

909. Calculate a) (−4) −3 b) 2,5

−1

 3 c)  −   4  1 d) 1   3

−2

e) −0, 4−4

−3

 1 f) −  2   2

910. Compare value of exponential with zero. b) 2, 6−4 c) (−7,1) −6 a) 9−5

−2

d) (−3,9) −3

911. Is it true? a) If a > 0 and n is integer than a n > 0 b) If a < 0 and n is even negative than a n > 0 c) If a < 0 and n is odd negative than a n < 0 912. Find value of x p if a) x = −7 and p = −2 b) x = 8 and p = −1

c) x = 2 and p = −6 d) x = −9 and p = 0

913. Find value of x p if a) x = −1 and p = −2 b) x = 0,5 and p = −2

c) x = 2 and p = −1 d) x = 0,5 and p = −5 67

914. Find value of x n and x − n if 2 a) x = and n = −2 3

b) x = −1,5 and n = 3

915. Find value of expression −1

a) 8 ⋅ 4

−3

b) −2 ⋅10−5 c) 18 ⋅ (−9) −1

 1 d) 10 ⋅  −   5 e) 3−2 + 4−1 f) 2−3 − (−2) −4

1 g) 0,5 +   3 0 h) 0,3 + 0,1−4

−1

−2

i) (−2,1)0 − (−0, 2) −2

916. Calculate −1

a) 6 ⋅12

−1

b) −4 ⋅ 8−2

−1

−2

c) 6 − 3

d) 1,30 − 1,3−1

1 e) 12 −   6 f) 25 + 0,1−2

917. Show expression as fraction without negative powers g) 2( x + y ) −4 c) 5ab −7 e) x −1c −3 a) 3x −5 b) x −4 y d) 5(ab) −7 f) −9 yz −8 h) 10 x −1 ( x − y ) −3 918. Show expression as fraction 1 3 2a 8 e) 2 3 a) 2 c) 5 x y b c x a5 ( a + b) 2 b) d) 3 f) y 7b b 4c 4 919. Show expression as fraction a) a −2 + b −2 b) xy −1 + xy −2

g)

2a (a − 2) 2

(c + b)5 h) 2(a − b) 4

c) (a + b −1 )(a −1 − b) d) ( x − 2 y −1 )( x −1 + 2 y )

920. Convert expression to fraction 68

a) (a −1 + b −1 )(a + b) −1

b) (a − b) −2 (a −2 + b −2 )

34. Properties of exponentials with integer powers If a ≠ 0 and m, n are integers then a m × a n = a m+ n a m : a n = a m−n (a m ) n = a mn If a ≠ 0 , b ≠ 0 and n is integer then (ab) n = a nb n a an ( )n = n b b

Examples: a −17 × a 21 = a −17 + 21 = a 4 b 2 : b5 = b 2−5 = b −3 1 (2a 3b −5 ) −2 = 2−2 (a 3 ) −2 (b −5 ) −2 = a −6b10 4

Exercises of Properties of exponentials with integer powers 925. Find value of expression a) 3−4 ⋅ 36 d) 210 ⋅ 212

g) (2−4 ) −1

b) 24 ⋅ 2−3

e) 5−3 ⋅ 5−3

h) (52 ) −2 ⋅ 53

c) 108 ⋅10−5 ⋅10−6

f) 3−4 ⋅ 3

i) 3−4 ⋅ (3−2 ) −4

926. Calculate a) 5−15 ⋅ 516

c) 4−8 : 4−9

e) (2−2 ) −3

−4

1 1 b)   ⋅    3  3

3

1 d)   5

2

1 :   3

4

f) (0,1−3 ) −1

927. Prove opposite powers of nonzero number are opposite. 69

a 928. Prove if a ≠ 0 and b ≠ 0 then   b

−n

b =  a

n

929. Calculate 1 a)   3

−3

3 b)   4

c) 0, 01−2 −1

 2 d) 1   3

e) 0, 002−1

−4

 1 f)  −1   2

−5

930. Prove if a and b are positive and a > b then a −1 < b −1 931. Show expression as power of 3 and calculate value of expression a) 27 ⋅ 3−4 c) 9−2 : 3−6 b) (3−1 )5 ⋅ 812 d) 813 : (9−2 ) −3 932. Show expression as power of 2 1 10 ⋅2 a) b) 32 ⋅ (2−4 ) 2 c) 8−1 ⋅ 43 16 933. Show expression as power of 5 b) (5m ) 2 ⋅ (5−3 ) m a) 5m ⋅ 5m +1 ⋅ 51− m

d) 45 ⋅16−2

c) 625 : 54 m− 2

934. Calculate a) 8−2 ⋅ 43

c) 100 :10−3

b) 9−6 ⋅ 275

d) 125−4 : 25−5

2−21 4−5 ⋅ 4−6 4−2 ⋅ 8−6 f) 2−22 e)

g)

3−10 ⋅ 98 (−3) 2

h)

5−5 ⋅ 2510 1253

935. Find value of expression 70

−1

a) 125 ⋅ 25

2

b) 16−3 ⋅ 46

c) (62 )6 : 614 d) 120 : (12−1 ) 2

(23 )5 ⋅ (2−6 ) e) 42 (3−2 )3 ⋅ 94 f) (33 ) 2

936. Show x −10 as product of powers in three different ways. 937. If a ≠ 0 then show a12 as power of a) a 4 b) a −6 938. Show power of x a) x10 : x12 b) x 0 : x −5

c) x n −1 : x −8 n is integer d) x 6 : x n + 2 n is integer

939. Simplify expression a) 1,5ab −3 ⋅ 6a −2b 3 −2 4 m n ⋅ 8m3n −2 4 1 c) 0, 6c 2 d 4 ⋅ c −3 d −4 3 b)

5 d) 3, 2 x −1 y −5 ⋅ xy 8 1 −1 −3 1 2 −5 e) p q ⋅ p q 2 6 1 5 −18 f) 3 a b ⋅ 0, 6a −1b 20 3

940. Find value of expression a) 0, 2a −2b 4 ⋅ 5a 3b −3 if a = −0,125 and b = 8 1 −1 −5 1 1 b) a b ⋅ 81a 2b 4 if a = and b = 27 7 14 941. Simplify expression and find value of expression a) 1, 6 x −1 y12 ⋅ 5 x 3 y −11 if x = −0, 2 and y = 0, 7 5 1 b) x −3 y 3 ⋅ 30 x3 y −4 if x = 127 and y = 6 5 942. Show power as product 71

−3

−1 −1 −2

−3 5 −12

a) (a b )

c) (0,5a b )

b) ( x 3 y −1 ) 2

d) (−2m5 n −3 ) 2

1  e)  p −2 q 2  3  −3 4 3 f) (−0,5 x y )

943. Convert to product 7  c)  p −6 q  8 

−5 −1 −1

a) (6a b )

3  b)  a −1b −3  4 

−1

2

d) (−0,3x −5 y 4 ) −2

944. Show expression as power of product a) 0, 0001x −4 c) 0, 0081a8b −12 b) 32 y −5

d) 10n x −2 n y 3n n is integer

945. Simplify expression 12 x −5 y a) −6 ⋅ 36 x −9 y

5 x −1 y 3 9 x 6 c) ⋅ −2 3 y

63a 2 18b 2 b) ⋅ 2b −5 7a 946. Evaluate expression 13 x −2 y12 ⋅ a) y 39 x −3 b)

5a 5 7b −3 ⋅ b −7 25a

d)

16 p −1q 2 25 p 6 ⋅ 5 64q −8

c)

p 15c ⋅ 3c −2 p −2

d)

26 x17 y ⋅ −8 13 x 25 y

947. Simplify expression

 x −3  a) (0, 25 x y ) ⋅  2   4y  −4

−3 2

−3

−2

 c −4  c)  ⋅ (5a 3bc 2 ) −2 5 2  10 a b   72

 a −3b 4   3  b)   ⋅  −2 3   9  a b 

−3

−3

 x 2 y −3   x 2 y −2  d)   ⋅   6z   9z 

2

948. Evaluate expression −2

 2 x −1  a)  −2  ⋅12 xy 5  3x   ab  b) 4a b ⋅    5 

−1

7 −1

a c) (2a b ) ⋅   b

−6

−2 3 2

−1

 2x2  d)  3  ⋅ ( x −1 y )3  y 

35. Scientific notation We write numbers in the form α ×10n where 1 ≤ α < 10 and n is integer. This form is scientific notation. Examples: 4350000 = 4,35 ×106 0, 000508 = 5, 08 ×10−4 Exercises

Error calculations 36. Writing approximately (with error) Example: l = 18 ∓ 0,3 means 18 − 0,3 < l < 18 + 0,3 17, 7 < l < 18,3 Example: p = 1, 429 here error is 0, 001 so we can show error with p = 1, 429 ∓ 0, 001 Example: m = 7,35 × 1022 , here error is (7,35 ∓ 0, 01) × 1022 = 7,35 ×1022 ∓ 0, 01× 1022 = 7,35 × 1022 ∓ 1020 so error is 1020 Relative error is the quotient of error : value = 7,35 × 1022 ∓ 1020 has error 1020 so relative error is 73

1020 1 = 22 7,35 × 10 735

Exercises

37. Operations on approximate value In addition and subtraction we write numbers in decimal forms and calculate result then write result in decimal form. But we count decimal digits in numbers and use minimum number of decimal digits in result. Example: x ≅ 17, 2 and y ≅ 8, 407 calculate x + y x + y ≅ 17, 2 + 8, 407 ≅ 25, 607 ≅ 25, 6 Example: x ≅ 6, 784 and y ≅ 4,91 calculate x − y x − y ≅ 6, 784 − 4,91 ≅ 1,874 ≅ 1,87 In multiplication and division we write numbers in scientific notation and again count number of decimal digits. Example: x ≅ 0,86 and y ≅ 27,1 calculate xy x ≅ 0,86 ≅ 8, 6 × 10−1 y ≅ 27,1 ≅ 2, 71× 101 x × y ≅ 8, 6 × 10−1 × 2, 71× 101 ≅ 2,3306 × 101 ≅ 2,3 ×101 ≅ 23 Example: x ≅ 563, 2 and y ≅ 32 calculate x : y x ≅ 563, 2 ≅ 5, 63 ×102 y ≅ 32 ≅ 3, 2 × 101 x : y ≅ (5, 63 × 102 ) : (3, 2 × 101 ) ≅ 1, 76 × 101 ≅ 1, 7 ×101 ≅ 17 Example: x ≅ 3, 75 and y ≅ 48,8 and z ≅ 0, 0095 calculate ( x + y) × z x + y ≅ 3, 75 + 48,8 ≅ 52,55 ≅ 52, 6 ( x + y ) × z ≅ 52, 6 × 0, 0095 52, 6 = 5, 26 × 101 0, 0095 = 9,5 ×10−3 74

52, 6 × 0, 0095 ≅ 5, 26 × 101 × 9,5 × 10−3 ≅ 4,997 × 10−3 ≅ 5, 0 × 10−1 ≅ 0,5 Exercises

Extra exercises for chapter V 1036. Find value of expression a) 10x −3 if x = 0,1

b) xy −4 if x = 200 and y = 5

1037. Prove values of expressions are opposite 4

3 a)   and 0, 6−4 4

c) 1000−2 and 0, 001−2

b) 1, 253 and 0,83

2 d) 2,5−4 and   5

−4

1038. Compare expressions a) 5

−3

and 7

−3

1 b)   2

−5

1 and    3

−5

c) (−2)0 and (−2) −2

1039. Calculate −2

a) −0, 25 ⋅100

c) 0, 2 ⋅ (1, 6)

1 2 e) 3 ⋅   − 0,5 3 3

b) 0, 01⋅ (−0,5) −3

d) 0,1−1 ⋅1,10

f) −4−1 ⋅ 5 + 2,52

−2

−4

1040. Change expression to a new expression without negative power (a + b)b 2a −1b 2 am −2 b) −1 c) a) −1 b ( a − b) (a + b) −2 a b 1041. Show expression as fraction a) xy −2 − x −2 y c) mn(n − m) 2 − n(m − n) −1 75

−1

x x b)   +    y  y

−2

d) ( x −1 + y −1 )( x −1 − y −1 )

1042. Simplify expression x −1 + y −1 a) ( x + y)2

ab −1 − a −1b b) a −1 − b −1

1043. Simplify expression a) 0,3a −2b3 ⋅1,5a 2b −1

d) (−0, 2m 2 n3 ) −3 ⋅ 0,1m6 n9

b) 6−1 x 2 y −1 ⋅1,5 xy −2

e) a −2b5 ⋅ (3ab) −1

c) 1, 2 xy −2 ⋅ 4 x −1 y

f) 6,1−3 y ⋅ (0,1xy −1 ) −1

1046. Show expression with one factor a) x b) x −1

c) x −2

1047. Remove factor given below outside of parenthesis from expression a −6 + a −4 b) a −6 a) a −4 1049. Prove equation is true with integer values of n a) 2n + 2n = 2n +1 b) 2 ⋅ 3n + 3n = 3n +1

76