Notes 6-1

Solving Inequalities: Addition and Subtraction

y 2x – 3

A. An inequality is a statement that two quantities are not equal. The quantities are compared by using the following signs:

A solution of an inequality is any value that makes the inequality true. The set of all solutions of an inequality is its solution set.

≤ A ≤ B

A is less than or

equal to B.

< A < B

A is less than B.

> A > B

A is greater than B.

≠ A ≠ B

A is not equal to B.

≥ A ≥ B

A is greater than or

equal to B.

I. Review: Inequalities

II. Graphing Inequalities on a Number Line A. How to Graph Inequalities

Graphing Inequalities

WORDS ALGEBRA GRAPH

All real numbers less than 5

All real numbers greater than -1

All real numbers less than or equal to 1/2

All real numbers greater than or equal to 0

–4 –3 –2 –1 0 1 2 3 4 5 6

–4 –3 –2 –1 0 1 2 3 4 5 6

–4 –3 –2 –1 0 1 2 3 4 5 6

–4 –3 –2 –1 0 1 2 3 4 5 6

1/2

x < 5

x > -1

x ≤ 1/2

x ≥ 0

Graph each inequality.

Ex 1: c > 2.5 Draw an empty circle at 2.5.

–4 –3 –2 –1 0 1 2 3 4 5 6

2.5

Graph each inequality.

Ex 2:. m ≤ –3 Draw a solid circle at –3.

Shade in all numbers less

than –3 and draw an arrow

pointing to the left.

–4 –2 0 2 4 6 8 10 12 –6 –8

−3

B. Examples

Shade in all the numbers greater

than 2.5 and draw an arrow pointing

to the right.

Write the inequality shown by each graph.

Use any variable. The arrow points to the left, so use either < or ≤. The empty circle at 2 means that 2 is not a solution, so use <.

x < 2

Use any variable. The arrow points to the right, so use either > or ≥. The solid circle at –0.5 means that –0.5 is a solution, so use ≥.

x ≥ –0.5

Ex. 1.

Ex. 2.

III. Writing Inequalities

You have seen that one way to show the solution set of an inequality is by using a graph. Another way is to use set-builder notation.

The set of all numbers x such that x has the given property.

{x : x < 6}

Read the above as “the set of all numbers x such that x is less than 6.”

IV. Set-Builder Notation

V. Solving Inequalities

Ex 1: Solve the inequality and graph the solutions.

x + 12 < 20

–12 –12

x + 0 < 8

x < 8

Since 12 is added to x,

subtract 12 from both sides

to undo the addition.

–10 –8 –6 –4 –2 0 2 4 6 8 10

Draw an empty circle at 8.

Shade all numbers less

than 8 and draw an arrow

pointing to the left.

A. With a few exceptions (which we will get to tomorrow), we solve inequalities in the same way we solve equations. Pretend the inequality symbol is an equal symbol.

Since there can be an infinite number of solutions to an inequality, it is not possible to check all the solutions. You can check the endpoint and the direction of the inequality symbol. Ex: The solutions of x + 9 < 15 are given by x < 6.

B. Checking Solutions

Caution! In Step 1, the endpoint should be a solution of the related equation, but it may or may not be a solution of the inequality.

d – 5 > –7 Since 5 is subtracted

from d, add 5 to both

sides to undo the

subtraction.

Draw an empty circle at –2.

Shade all numbers

greater than –2 and

draw an arrow pointing

to the right.

+5 +5 d + 0 > –2

d > –2

Ex 2: Solve the inequality and graph the solutions.

–10 –8 –6 –4 –2 0 2 4 6 8 10

Ex 3: Solve the inequality and graph the solutions.

0.9 ≥ n – 0.3 Since 0.3 is subtracted

from n, add 0.3 to both

sides to undo the

subtraction.

Draw a solid circle at 1.2.

Shade all numbers

less than 1.2 and draw

an arrow pointing to

the left.

0 1 2

+0.3 +0.3 1.2 ≥ n – 0

1.2 ≥ n

1.2

a. s + 1 ≤ 10

–1 –1 s + 0 ≤ 9

s ≤ 9

Since 1 is added to s, subtract 1 from

both sides to undo the addition.

b. > –3 + t Since –3 is added to t, add 3 to both

sides to undo the addition.

Ex 4: Solve each inequality and graph the solutions.

+3 +3

> 0 + t

t <

9 –10 –8 –6 –4 –2 0 2 4 6 8 10

–10 –8 –6 –4 –2 0 2 4 6 8 10

VI. Reading Math

Common Phrase Equivalent Phrase Symbol

No more than

At most

No less than

At least

Less than or equal to

Less than or equal to

Greater than or equal to

Greater than or equal to

≤

≤

≥

≥

Ex. 1. Ray’s dad told him not to turn on the air conditioner unless the temperature is at least 85°F. Define a variable and write an inequality for the temperatures at which Ray can turn on the air conditioner. Graph the solutions.

Let t represent the temperatures at which Ray can turn on the air conditioner.

75 80 85 90 70

Turn on the AC when temperature is at least 85°F

t ≥ 85

Draw a solid circle at 85. Shade all

numbers greater than 85 and draw an

arrow pointing to the right. t 85

Ex. 2: A store’s employees earn at least $8.25 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions.

Let d represent the amount an employee can earn per hour.

An employee earns at least $8.25

d ≥ 8.25

4 6 8 10 12 −2 0 2 14 16 18

8.25

d ≥ 8.25

4. A certain restaurant has room for 120 customers. On one night, there are 72 customers dining. Write and solve an inequality to show how many more people can eat at the restaurant. x + 72 ≤ 120; x ≤ 48, where x is a natural number

Lesson Quiz: Part I

1. Describe the solutions of 7 < x + 4.

all real numbers greater than 3

2. Graph h ≥ –4.75

–5 –4.75 –4.5

Write the inequality shown by each graph.

3. x ≥ 3

4. x < –5.5

Lesson Quiz: Part II

5. A cell phone plan offers free minutes for no more than 250 minutes per month. Define a variable and write an inequality for the possible number of free minutes. Graph the solution.

0 ≤ m ≤ 250

0 250

Let m = number of minutes.

Lesson Quiz: Part III

Solve each inequality and graph the solutions.

1. 13 < x + 7

x > 6

2. –6 + h ≥ 15

h ≥ 21

3. 6.7 + y ≤ –2.1

y ≤ –8.8