Equations & Inequalities -i

EQUATIONS & INEQUALITIES-I EQUATION An equation is a statement that two expressions are equal, e.g. 3x @ 9 = 6, 2x + y = 11, 2x 2 @ x @ 1 = 0, x 2 + 2 y 2 = 11 etc It may contain one or more variables. An equation containing variables is neither true nor false. It is true for certain values, and false for others. SOLUTION : The equation 3x – 9 = 6, is true for x = 5. If we put 5 in place of x in the equation the two sides become same. This is called a solution of the equation. The equation 2x+y = 11 has infinite solutions (one of them is x=2, y=7). 2 2 (x=3,y=1) is a solution of the equation x 2 + 2 y 2 = 11 as 3 + 2.1 = 11

The set of all solutions is called the solution set of the equation. The equation, 2x – 6 = 0 has a solution set { 3 }. The equation x 2 = 9 has a solution set { -3, 3 }.

Two equations are equivalent if they have the same solution set. PQ xf f f 2x @ 6 = 0 and x + 3 = f + 2 are equivalent equations as both have the solution set 3 A 3 Identity : An equation which is true for all value of possible variables, is called an identity.

Example : a + b = a 2 + 2ab + b is true for all real numbers a and b, so it is an identity A `

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2

Process of solving an equation : There are a number of procedures to be used to solve equations. It can be summarized as : 1. For every real number a, for every real number b, for every real number c, a=b if and only if a + c = b` + c a a=b if and only if ac = bc c ≠ 0 a=b if and only if a - c = b - c ` a af f f f bf f f a=b if and only if = f c≠0 c c 2. Zero factor property : For a and b are real number and ab = 0, then a = 0 or b = 0.

Linear Equation of one variable :

A linear equation of one variable has the form ax + b = 0, a ≠ 0 . It has exactly one solution. An equation which is not in this form but can be transformed into this form is also a linear equation of one variable. xf f f Thus x + 3 = f + 2 is a linear equation A It can be transformed as 3 3f f 2x + 3 = 0 which is in the form ax + b = 0 and has only one solution x = @ f 2 An equation which is not linear is called nonlinear. Example : 2y+10=0 is an example of a linear equation in one variable. It has one solution i.e. y = -5. The solution set is { -5 }. Solving Linear Equation of One Variable:

Linear equations can be solved by isolating the variable on one side A xf f f + 2A Example 1:solve the equation x + 3 = f 3 xf f f +2 multiply both sides by 3 x+3= f 3 3x + 9 = x + 6 subtract x from both sides 2x + 9 = 6 subtract 9 from both sides 2x = @ 3 divide both sides by 2 3f f x =@ f 2

3f f Solution set is @ f 2 V

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Quadratic equation of one variable: A quadratic equation of one variable is one that is in the form ax 2 + bx + c = 0 a ≠ 0 , where a, b, c are constants and x is the variable. This is its standard form. `

a

Note that this has only one variable x here. If the equation is given in any other form, then it can be changed into this form.

Solving Quadratic Equation of One Variable:

There are four methods of solving the quadratic equations of one variable : 1. Factoring : If the polynomial of the form ax 2 + bx + c can be factored in two linear factors, we have to write it in the factored form, apply the zero factor property that For a and b are real number and ab = 0, then a = 0 or b = 0. w w w w w w w

2. Square root property : If the equation is in the form a 2 = b, then a = F p b

3. Completing the square: We can write the equation in the form x 2 + px = q . To 2 pf f f f f f f on both sides. The we apply the complete the square on the left, we then add f 4 square root property as above. 4. Quadratic formula: The solutions of ax 2 + bx + c = 0 a ≠ 0 can be written as : x=

w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 2 q @ bf F bf @ 4ac f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f

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. This formula is actually derived by completing 2a the square on both sides. 2 The quantity, b @ 4ac is called the discriminant. The sign of this tells us about the nature of the solutions : 2

If b @ 4ac > 0, we have 2 real solutions A If b @ 4ac = 0, we have 2 real solutions which are equal, iε A 1 repeated real solution A 2 2

If b @ 4ac < 0, we have 2 imaginary solutions which are conjugate to each other A Usually, we first try to factorize it, if it is easily factorable. Other wise quadratic formula is used. Example 2 : Solve by factoring 2x 2 + 5x + 3 = 0 . Answer :

2x 2 + 5x + 3 = 0 The polynomial is easily factorable using integers ` a` a 2x + 3 x + 1 = 0 Apply the zoro @ factor property A 2x + 3 = 0 or x + 1 = 0 3f f f or x = @ 1 x =@ 2

Example 3 : Solve by completing the square 2x 2 + 5x + 4 = 0 .

Answer : 2x 2 + 5x + 4 = 0

The polynomial is not factorable, subtract 3 from both sides A

2x 2 + 5x = @ 4 divide both sides by 2 to get in the form x 2 + px = q 2 pf 5f f f f f f f f x = @ 2 add f on both sides x2 + f 4 2 5f 25 f 25 f f f f f f f f f f f f x+ f =@2 + f write the square form on the left x2 + f 2 16 16 f

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5f 7f f f f f f f f x+ f =@ f apply square root on both sides 4 16 w w w w w w w w w w w w w w w w w w w w w w w

5f f f s 7 f f f f f f x+ f =F @ f 4 16 w w w w w w w

p if 5f F 7f f f f f f f f f f f f f f f f f f f f f f f f f x= f 4

Example 3 : Solve by quadratic formula 2x 2 + 3x @ 4 = 0 .

Answer : 2x 2 + 3x @ 4 = 0 The polynomial is not factorable Comparing the equation with standard form we get a = 2, b = 3, c = @ 4 Using quadratic formula we get, w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w `w a 2

q3 @ 4.2 A @ 4 @f 3f F f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f x= f 2.2 w w w w w w w w w w w w p @ 3f F 41 f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f = 4

INEQUALITIES An inequality is a statement that two expressions are not equal, i.e. one expression is either ‘greater than (>)’ or ‘greater than or equal to ( ≥ )’ or ‘less than (<)’ or ‘less than or equal to ( ≤ )’ the other expression. It may contain one or more variables. Some examples of inequalities are : 3x @ 9 > 6, 2x + y ≥ 11, 2x 2 @ x @ 1 < 0,

x 2 + 2 y 2 ≤ 11 etc

Like an equation, an inequality statement containing variables is neither true nor false. It is true for certain set of values, and false for others. For inequality of one variable, a value of the variable that makes the statement true is a solution to the inequality. The set of all solutions is called the solution set of the inequality. Two inequalities are equivalent, if they have the same solution set. For example : 2x < b c 10 and x-5 < 0 are equivalent inequalities as both have the solution set @1 , 5 . Here we shall discuss inequality statement of one variable only.

Solving an Inequality To solve inequalities we transform an inequality to an equivalent inequality, by performing some operations on it using the properties of inequality (refer to the ‘Properties of Inequality’ discussed in the chapter : Sets, Numbers, Operations, Properties) and then simplifying expressions on both sides of the inequality.

Linear Inequality A linear inequality is in the form ax + b>0, ax + b<0, ax + b ≥ 0, ax + b ≤ 0 or can be transformed to an equivalent inequality in this form. Solving Linear Inequality of one variable:

They are solved by isolating the variable to one side (similar to the process of solving linear equation of one variable).

Eaxmple : Solve the inequality 2x @ 5 ≤ 11 . Solution : 2x @ 5 ≤ 11 2x < 16 x <8

add 5 to both sides divide both sides by 2

Nonlinear Inequality An inequality of one variable, where we can simplify it to that extent, that we get 0 on one side, and the other side can be written in the factor form, can be solved through sign diagram. As example, the inequality x 2 < 2 @ x can be simplified as : x2 < 2 @ x x 2 + x a@` 2 < 0a ` x @1 x + 2 <0 which can be solved the following way. •

Determine the values of x where the factor becomes 0. These are called critical points .

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Draw a number line and plot the critical points on it, which divides the number lines in intervals.

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Determine the sign of each factor in each interval and thus determine the sign of the whole expression on the left side of the inequality.

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Write the solution set.

Example : Solve : x 2 < 6 + x Solution : x2 < 6 + x x 2 @ x a@` 6 < 0a ` x @2 x + 3 <0 making x @ 2 = 0 we get x = 2; and from x + 3 = 0 we get x = @ 3; The critical points are + 2 and @ 3 which divides the number line in the intervals @1 , @ 3 b

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c

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@ 3,2 and 2, 1 A `

@ ve | @ ve | + ve @ ve | + ve | + ve + ve | @ ve | + ve P @@ | @@@ | @@@ | @@@ | @@@ | @@@ | @@@Q @3 @2 @1 0 1 2

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Sign of x @ 2 ` a Sign of x + 3 ` a` a Sign of x @ 2 x + 3 the number line

From the above we see that in the interval @ 3,2 , x @ 2 is negative but x + 3 is positive, b

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thus the product x @ 2 x + 3 is negaive which satisfies x @ 2 x + 3 <0 `

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b

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So the solution set is the interval @ 3,2 A

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