Buksis-4.4 Revisi Akhir

4.4 Definition of Linear Inequality with One Variable

What are you going to learn? To define inequality To define a linear inequality with one variable

Key Terms: • • •

inequality linear Inequality with one variable solution and solution set

Consider the number of students in your class. How many students are in your class?

If the sentence “The number of students in this class is less than 25 persons” is grouped according to the phrases, we get English expression

Mathematical Expression

Number of students in this class

Is less than

25

Let n be the number of students in this class, then n < 25.

Now look at Figure 4.14 below.

(i)

Max 60 Km

(ii) 17 years

(iii)

(iv) Maximum Passengers 6 people

Passengers cannot exceed 15 people

Figure 4.14

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Look at Figure 4.14 that describes a real life situation.

i) it means that maximum speed is 60 km/hr. ii) it means that the person to watch the movie must be 17 years old or more. iii) it means that the number of passengers of the car cannot be more than 6. iv) it means that the maximum number of the passengers is 15. Work in groups or pairs. Using Figure 4.14. Answer the following questions 1. Express your opinion, why is there a rule in each figure above? 2. Let t be the speed of a car, m be the age of visitors, s be the number of passengers of the car, h be the number of passengers of a ship. Write the condition of t, m, s and h in the mathematical expression.

Problem 3

Look again at your answer to Problem 2. a). Does each requirement that you have written has a variable? b). How many variables are there in each requirement? c). What is the power of the variable? d). Which notation do you use in your answer to Problem 2? ( “=”, “≤” , “≥”, “<”, “>” ) e).

In your answers to problem 2, which ones are open

sentences?

An open sentence using the sign “>”, “≥“ , “<”, or “≤” is called an inequality. An inequality that contains one variable of which the power is one is called a linear inequality with one variable.

148 / Student’s Book – Linear Equations and Inequalities with One Variable

From your answer to problem 2, which sentence is called a linear inequality with one variable?

Figure 4.14 gives some examples of a real life situation related to the linear inequality with one variable. Find another example of daily life situation related to weight, height, square, volume, report grades or others which can be stated in a linear inequality with one variable.

Problem 4

Ida has 5 packs of writing books. Diah has 3 packs of writing books. The number of writing books in each pack is the same. Ida gives 3 books to Susi and Diah receives extra 9 books from her mother. The number of Diah‘s books is more than the number of Ida’s books. If each pack contains n pieces of books, a).

write the relation between 5n – 3 and 3n + 9.

b).

find the value of n so that it holds for that relation.

c).

find the value of n so that it does not hold for that

relation.

Every inequality contains variables. The substitution of a variable that makes the sentence true is called a solution of the inequality. The set of all solutions is called a solution set of that inequality. -5 is a solution of the inequality 2x – 5 < -x + 2, because 2.(-5) – 5 < -(-5) + 2 is a true statement. 4 is not a solution 4.(4) – 12 ³ of the inequality 4t – 12 ³ > 2t + 1, because >2.(4) + 1 is a false statement.

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Solving A Linear Inequality with One Variable Sketching the graph of solution in a line number Look at the following line number and then answer the questions below. •

-5

•

-4

•

-3

•

-2

•

-1

•

0

• 1

•

2

•

3

• 4

•

5

What numbers are solutions of the inequality x < 3? Is 4 a solution of that inequality? Is 3 a solution of that inequality? Is 2 a solution of that inequality? Is 1 a solution of that inequality? Is 0 a solution of that inequality? Is -1 a solution of that inequality? Is -2 a solution of that inequality? Is -3 a solution of that inequality? Can you mention all solutions of that inequality? The solutions can be described on the following number line.

•

-5

•

-4

•

-3

•

-2

•

-1

•

0

• 1

•

2

⏐ 0 3

x = 3 on the line is not dotted because 3 is not a solution. The graph of solution of t ≤ 3 is

150 / Student’s Book – Linear Equations and Inequalities with One Variable

•

•

•

-4

-5

•

-3

•

-2

•

•

0

-1

⏐ •

•

1

2

3

x = 3 in the graph is dotted, because 3 is also a solution.

Problem 5

Sketch the graph solution of the following inequality on a number line. a. y ≥ -1

c. n ≤ 0.

b. m < 5

Working out an Inequality by Addition or Division Look at statement -4 < 1. That statement is true. The number line below shows what happens if 2 is added to both sides. +2

+2

•

-5

•

-4

•

-3

•

-2

•

-1

•

0

• 1

•

2

•

3

•

4

•

5

If both sides are added by 2, then we obtain a statement -2 < 3. That statement is also true. In the example above, adding 2 to both sides does not change the truth value of the statement. Now, look at statement -3 < 1. That statement is true.

The line number below shows what happens if 2 is subtracted from both sides.

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-2

-2

•

-5

•

-4

•

-3

•

-2

•

-1

•

0

• 1

•

•

2

3

• 4

•

5

If 2 is subtracted from both sides, then we obtain a statement -5 < -1. That statement is still true. In the above example, subtracting 2 from both of sides does not change the truth of the statement.

Add or subtract a certain number as you wish from both sides. Are the statements that you have always true?

Properties of addition or subtraction in an inequality If a certain number is added to or subtracted from both sides of an inequality, the symbol of the inequality does not change, and the solution does not change, either. The new linear inequality that we get if a certain number is added to or subtracted from both sides is called a linear inequality equivalent to the original one.

Example 1

Find the solution set of the following inequalities: a. y + 2 > 6 b. x – 3 ≤ 2, x is an integer between −3 and 8. Solution : a. y + 2 > 6 ⇔ y + 2 – 2 > 6 –2 both sides) ⇔ y>4 The graph :

( 2 is subtracted from

152 / Student’s Book – Linear Equations and Inequalities with One Variable

⏐ 0 4

•

5

b. x – 3 ≤ 2 ⇔

x – 3 + 3 ≤ 2 + 3

( 3 is subtracted from

both sides) ⇔

x≤5

Another way: Because the solutions are not so many, we can check them one by one. x = -2 ⇒ (-2) – 3 ≤ 2 x = 3 ⇒ (3) – 3 ≤ 2 -5 ≤ 2 (true) 0 ≤ 2 (true) x = -1 ⇒ (-1) – 3 ≤ 2 x = 4 ⇒ (4) – 3 ≤ 2 -4 ≤ 2 (true) 1 ≤ 2 (true) x = 0 ⇒ (0) – 3 ≤ 2 x = 5 ⇒ (5) – 3 ≤ 2 -3 ≤ 2 (true) 2 ≤ 2 (true) x = 1 ⇒ (1) – 3 ≤ 2 x = 6 ⇒ (6) – 3 ≤ 2 -2 ≤ 2 (true) 3 ≤ 2 (false) x = 2 ⇒ (2) – 3 ≤ 2 x = 7 ⇒ (7) – 3 ≤ 2 -1 ≤ 2 (true) 4 ≤ 2 (false) Thus, the solution is -2, -1, 0, 1, 2, 3, 4, 5

In your opinion, which way is easier and more efficient? Comprehension Check Find the solution set and sketch the graph of the solution of the following inequalities. a. w + 2 > -1 b. 8 < 5 + r 3

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Working out Inequality by Multiplication or Division Work in groups. Consider the statement 4 > 1 and the statement 8 < 12. Those two statements are true. Fill in the blanks below. First fill it with a suitable number, and then fill it with the sign “<“, “>“ or “= “.

4>1 12 = 4 . 3

1. 3

=

3

(both sides are

1. 2

=

. . .

(both sides are

multiplied by 3) ... = 4.2 multiplied by 2) ... = 4.1

1. 1 = . . .(both sides are multiplied by 1)

... = 4.0

1. 0 = . . (both sides are multiplied by 0)

. . . = 4 . -1

1. -1 = . (both sides are multiplied by -1)

-8

1. -2 = -2 (both sides are multiplied by

= 4 . -2

-2) . . . = 4 . -3

1. -3 = . . (both sides are multiplied by -3)

8 < 12 ... = 8:4 4 = 8:2 ... = 8:

1 2

-8 = 8 : -1

12 : 4 = . . . (both sides are divided by 4) 12 : 2 = 6 12 :

(both sides are divided by 2)

1 1 = . . (both sides are divided by ) 2 2

12 : -1 = -12

(both sides are

divided by -1) . . . = 8 : -2

12 : -2 = . . (both sides are divided by -2)

154 / Student’s Book – Linear Equations and Inequalities with One Variable

. . . = 8 : -4

12 : -4 = . . .(both sides are divided by -4)

Compare the sign in the box that you have filled with the sign of the beginning statement. What happens if both sides are multiplied

by a positive number, by

zero, or by a negative number? And what happens if both sides are divided by a positive number, or by a negative number?

Properties of multiplication or division on both sides of an inequality On an inequality: 1. if both sides are multiplied or divided by a positive number (non zero), then the sign of the inequality does not change. 2. if both sides are multiplied or divided by a negative number (non zero), then the sign of the inequality changes into the opposite.

Example 2

Find the solution set of the following inequalities, and then sketch the graph of the solution on a number line. a. x < -1. 2 b. - 2 x ≥ 2. 3 c. 4x – 2 < -2x + 10, x is an integer between -1 and 8 Solution : a. x < -1 2 ⇔ 2.

x < 2. –1 2

(both sides are multiplied by 2, the

sign does not change) ⇔ x < -2.

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The graph :

•

-5

b.

•

-4

•

-3

⏐ 0

-2

- 2 x ≥ 2. 3 ⇔ 3.(- 2 x) ≥ 3.2 (both sides are multiplied by 3, 3 the sign does not change) ⇔ -2x ≥ 6 ⇔ − 2x ≤ 6 −2 −2

(both sides are divided by –2, the sign changes into the opposite)

⇔ x ≤ -3.

The graph : •

-5

•

-4

⏐ •

-3

A car can carry loads not more than 2000 kg. The weight of the driver and his assistant is 150 kg. He will lift some boxes of goods. The weight of each box is 50 kg. a) What is the maximum number of boxes that can be carried in one route? b) If he carries 350 boxes, what is the minimum number of the route that must be done?

156 / Student’s Book – Linear Equations and Inequalities with One Variable

1. Write an inequality that can state the following cases. a) The driver must be 17 years old or more. b) There are more than 20 species of crocodile. c) Bus passengers cannot exceed 60 people. 2. Which of the following statements is a linear inequality with one variable? If the statement is not true, give your reason. a)

–3t + 7 ≥ t

c) 2m – m < 0

b)

y . (y +2) > 2y – 1

d)

c) x – x2 > 3

y+y≤5

3. Find the solution set for each inequality, and then sketch the graph of the solution on a number line. 47 7 ≤t5 2

a) x – 1 > 10

f)

b) w + 4 ≤ 9

g) h -

c) –5 > b – 1

h) -7

d)

3 + k ≥ -45 2

e) 2 < s – 8

1 ≥ -1 2

3 1 1 + m + ≤ -2 4 2 4

i) –3.(v – 3) ≥ 5 – 3v j)

4 2 1 r–3
4. Critical Thinking Find the value of a so that the inequality ax + 4 ≤ -12 has the solution presented in the graph below.

•

-5

•

-4

•

-3

⏐ •

-2

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158 / Student’s Book – Linear Equations and Inequalities with One Variable