linier inequality with 1 variab

8/4/2019 Linier Inequality With 1 Variab

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at are you going to learn? To define ineq ua lity

To d efine a linea r

inequality w ith o ne

variable

Key Terms:

• inequality

• linea r Ineq ua lity w ith

one va riab le

• solution and

solution set

Definition of Linear Inequality withOne Variable

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

Let n be the number of students in this

class, then n < 25.

Now look at Figure 4.14 below.

Englishexpression

Number ofstudents inthis class

Is less than 25

44..44

MathematicalExpression

Max60

Km i

(iii)

(ii)

(iv)

17 years

Maximum Passengers6 people

Passengers cannot exceed

15 eo leFigure 4.14

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



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 questions1. 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 numberof 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.

Look again at your answer to Problem 2.Problem 3

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 isone is called a linear inequality with one variable.

Mathematics for Junior High School – Year 7/149



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.

Ida has 5 packs of writing books. Diah has 3 packs ofwriting 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.

Problem 4

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, because2.(-5) – 5 < -(-5) + 2 is a true statement. 4 is not a solution

of the inequality 4t – 12³ > 2t + 1, because 4.(4) – 12³

>2.(4) + 1 is a false statement.




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.

1• •

0• 2

• -1-3

• • -4

• -2

• -5 4

• • 3

• 5

What numbers are solutions of the inequality x < 3?

Is 4 a solution of that inequality?





Is -1 a solution of that inequality?



Can you mention all solutions of that inequality?

The solutions can be described on the following number

line.

1• •

0• 2

• -1-3

• • -4

• -2

• -5

0 3

⏐

x = 3 on the line is not dotted because 3 is not a solution.

The graph of solution of t ≤

3 is




1• •

0• 2

• -1-3

• • -4

• -2

• -5

• 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 b. m < 5 c. n ≤ 0.

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.

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.

1• •

0• 2

• -1-3

• • -4

• -2

• -5 4

• • 3

• 5

+2 +2




1• •

0• 2

• -1-3

• • -4

• -2

• -5 4

• • 3

• 5

-2 -2

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, andthe solution does not change, either.The new linear inequality that we get if a certain number isadded to or subtracted from both sides is called a linearinequality equivalent to the original one.

Find the solution set of the followinginequalities:

Example 1

a. y + 2 > 6b. x – 3 ≤ 2, x is an integer between −3 and 8.

Solution :a. y + 2 > 6

⇔ y + 2 – 2 > 6 – 2 ( 2 is subtracted fromboth sides)

⇔ y > 4The graph :




40 •

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 <3

5 + r




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

multiplied by 3)

. . . = 4 . 2 1. 2 = . . . (both sides are

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 = 4 . -2 1. -2 = -2 (both sides are multiplied by

-2)

. . . = 4 . -3 1. -3 = . . (both sides are multiplied by

-3)

8 < 12

. . . = 8 : 4 12 : 4 = . . . (both sides are divided by 4)

4 = 8 : 2 12 : 2 = 6 (both sides are divided by 2)

. . . = 8 :2

1 12 :

2

1= . . (both sides are divided by

2

1)

-8 = 8 : -1 12 : -1 = -12 (both sides are

divided by -1)

. . . = 8 : -2 12 : -2 = . . (both sides are divided by -2)




. . . = 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 bothsides of 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.

On an inequality:

Example 2 Find the solution set of the following inequalities, and

then sketch the graph of the solution on a number line.

a.2

x < -1.

b. -3

2 x ≥ 2.

c. 4x – 2 < -2x + 10, x is an integer between -1 and 8

Solution :

a.2

x < -1

⇔ 2.2

x < 2. –1 (both sides are multiplied by 2, the

sign does not change)

⇔ x < -2.




The graph :

-3• •

-40

-2•

-5

⏐

b. -3

2 x ≥ 2.

⇔ 3.(-3

2 x) ≥ 3.2 (both sides are multiplied by 3,

the sign does not change)

⇔ -2x ≥ 6

⇔ 22−−

x ≤ 26− (both sides are divided by –2, the

sign changes into the opposite)

⇔ x ≤ -3.

The graph :

-3• •

-4• -5

⏐

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?




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 c) x – x2 > 3

b) y . (y +2) > 2y – 1 d) y + y ≤

53. Find the solution set for each inequality, and then sketch the

graph of the solution on a number line.

a) x – 1 > 10 f)5

47 ≤ t -

2

7

b) w + 4 ≤ 9 g) h -2

1 ≥ -1

c) –5 > b – 1 h) -74

3 + m +2

1 ≤ -24

1

d)2

3+ k ≥ -45 i) –3.(v – 3) ≥ 5 – 3v

e) 2 < s – 8 j)3

4r – 3 < r +

3

2-3

1r

4. Critical Thinking Find the value of a so that the

inequality ax + 4 ≤ -12 has the solution presented in

the graph below.

-3• •

-4•

-2•

-5

⏐


linier inequality with 1 variab

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