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California Standards. AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Vocabulary. inequality

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California

Standards

AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.


Vocabulary

inequality

algebraic inequality

solution set


An inequalitycompares two expressions using <, >, , or .

is less than

Fewer than, below

is greater than

More than, above

is less than or equal to

At most, no more than

is greater than or equal to

At least, no less than

An inequality that contains a variable is an algebraic inequality.


A solution of an inequality is any value of the variable that makes the inequality true. All of the solutions of an inequality are called the solution set.

You can graph the solution set on a number line. The symbols < and > indicate an open circle.


This open circle shows that 5 is not a solution. that makes the inequality true. All of the solutions of an inequality are called the

a > 5

The symbols ≤ and ≥ indicate a closed circle.

This closed circle shows that 3 is a solution.

b ≤ 3


Draw a closed circle that makes the inequality true. All of the solutions of an inequality are called the

at –2 and all values of z greater than -2 . So shade to the right of –2 .

1 2

1 2

1 2

Additional Example 3: Graphing Inequalities

Graph each inequality.

A. –1> y

Draw an open circle at –1. The solutions are all values of y less than –1, so shade the line to the left of –1.

–3 –2 –1 0 1 2 3

12

B. z ≥ –2

–3 –2 –1 0 1 2 3


Check It Out! that makes the inequality true. All of the solutions of an inequality are called the Example 3

Graph each inequality.

A. n < 3

Draw an open circle at 3. The solutions are all values of n less than 3, so shade the line to the left of 3.

–3 –2 –1 0 1 2 3

B. a ≥ –4

Draw a closed circle

at –4. The solutions are all values greater than –4, so shade to the right of –4.

–6 –4 –2 0 2 4 6


When you solved two-step equations, you used the order of operations in reverse to isolate the variable. You can use the same process when solving two-step inequalities.


When you multiply (or divide) both sides of an inequality by a negative number, you must reverse the inequality symbol to make the statement true.


1 2 3 4 5 6 7 a negative number, you must reverse the inequality symbol to make the statement true.

12

4x

>

4

4

Example 1: Solving Two-Step Inequalities

Solve and graph.

4x + 1 > 13

4x + 1 > 13

– 1– 1Since 1 is added to 4x, subtract 1 from both sides.

4x > 12

Since x is multiplied by 4, divide both sides by 4.

x > 3


Example 1 Continued a negative number, you must reverse the inequality symbol to make the statement true.

Check

According to the graph 4 should be a solution and 2 should not be a solution.

x > 3

x > 3

Substitute 4 for x.

Substitute 2 for x.

?

?

4 > 3

2 > 3

So 4 is a solution.

So 2 is not a solution.


a negative number, you must reverse the inequality symbol to make the statement true.

-6 -5 -4 -3 -2 -1 0

18

–9x

–9

–9

Example 2: Solving Two-Step Inequalities

Solve and graph.

–9x + 7  25

–9x + 7  25

– 7– 7Subtract 7 from both sides.

–9x 18

Divide each side by –9; change  to .

x–2


1 2 3 4 5 6 7 a negative number, you must reverse the inequality symbol to make the statement true.

10

5x

>

5

5

Check It Out! Example 3

Solve and graph.

5x + 2 > 12

5x + 2 > 12

– 2– 2Subtract 2 from both sides.

5x > 10

Divide both sides by 5.

x > 2


Check It Out! a negative number, you must reverse the inequality symbol to make the statement true. Example 3 Continued

Check

According to the graph 4 should be a solution and 1 should not be a solution.

x > 2

x > 2

Substitute 4 for x.

Substitute 1 for x.

?

?

4 > 2

1 > 2

So 4 is a solution.

So 1 is not a solution.


a negative number, you must reverse the inequality symbol to make the statement true.

-6 -5 -4 -3 -2 -1 0

16

–4x

–4

–4

Check It Out! Example 4

–4x + 2  18

–4x + 2  18

– 2– 2Subtract 2 from both sides.

–4x 16

Divide each side by –4; change  to .

x–4


Example 5 a negative number, you must reverse the inequality symbol to make the statement true.

<

10

+

4y

18

<

Simplify.

4y

8

Divide each side by 4.

4y

8

<

4

4

<

Simplify.

y

2

0

1

2

3

4

5

Solving and Graphing a Two-Step Inequality

Original inequality

<

10

+

4y

Subtract 10 from each side.

18

-10 -10


Example 6 a negative number, you must reverse the inequality symbol to make the statement true.

Multiple Choice Practice

x

15

x

15

x

25

x

25

SOLUTION

To find your profit, subtract the total costs from the total ticket sales. This amount should equal or exceed the minimum desired profit of $200.

You are organizing a bowling night for charity. Each ticket costs $10 and includes shoe rental. Door

prizes cost you $50. Which inequality describes the possible numbers x of people who need to attend for you to make a profit of at least $200?


Example 6 a negative number, you must reverse the inequality symbol to make the statement true.

Multiple Choice Practice

Write an inequality.

10x

50

200

10x

x

250

25

ANSWER

The correct answer is D.

+50 +50

Add 50 to each side.

Divide each side by 10.

10 10


Example 7 a negative number, you must reverse the inequality symbol to make the statement true.

<

4x

Combine like terms.

4

4

4

4x

14

14

Divide each side by and

reverse the inequality symbol.

>

7

x

>

Simplify.

2

0

1

2

3

4

5

Combining Like Terms

<

x

Original inequality

3x

14

<

Subtract 3x from each side.

3x

14

x

3x

3x


Guided Practice a negative number, you must reverse the inequality symbol to make the statement true.

1.

ANSWER

57

z

7z

15

+

6

<

2.

<

ANSWER

n

+

5

11n

40

3n

Solve the inequality. Then graph the solution.


Guided Practice a negative number, you must reverse the inequality symbol to make the statement true.

Solve the inequality. Then graph the solution.

2

3.

ANSWER

y

16

9y

18

>

>

9

for Examples 1, 2, and 3


Guided Practice a negative number, you must reverse the inequality symbol to make the statement true.

4. WHAT IF?

In Example 3, how many people need to attend for you to make a profit of at least $250?

x

ANSWER

30 people

for Examples 1, 2, and 3


Example 8: a negative number, you must reverse the inequality symbol to make the statement true. School Application

A school’s Spanish club is selling bumper stickers. They bought 100 bumper stickers for $55, and they have to give the company 15 cents for every sticker sold. If they plan to sell each bumper sticker for $1.25, how many do they have to sell to make a profit?

In order for the Spanish club to make a profit, the revenue must be greater than the cost.

1.25x > 55 + 0.15x


55 a negative number, you must reverse the inequality symbol to make the statement true.

>

1.10x

1.10

1.10

Example 8 Continued

1.25x > 55 + 0.15x

Subtract 0.15x from both sides.

– 0.15x– 0.15x

1.10x > 55

Divide both sides by 1.10.

x > 50

The Spanish club must sell more than 50 bumper stickers to make a profit.


Check It Out! a negative number, you must reverse the inequality symbol to make the statement true. Example 9

A school’s French club is selling bumper stickers. They bought 200 bumper stickers for $45, and they have to give the company 25 cents for every sticker sold. If they plan to sell each bumper sticker for $2.50, how many do they have to sell to make a profit?

In order for the French club to make a profit, the revenue must be greater than the cost.

2.5x > 45 + 0.25x


45 a negative number, you must reverse the inequality symbol to make the statement true.

>

2.25x

2.25

2.25

Check It Out! Example 9 Continued

2.5x > 45 + 0.25x

Subtract 0.25x from both sides.

– 0.25x– 0.25x

2.25x > 45

Divide both sides by 2.25.

x > 20

The French club must sell more than 20 bumper stickers to make a profit.


1 2 3 4 5 6 7 a negative number, you must reverse the inequality symbol to make the statement true.

-10 -9 -8 -7 -6 -5 -4

-18 -17 -16 -15 -14 -13 -12

Lesson Quiz: Part I

Solve and graph.

1. 4x – 6 > 10

2. 7x + 9 < 3x – 15

3.w – 3w < 32

x > 4

x < –6

w > –16


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