Warm UpSolve each inequality. 1. x + 3 ≤ 10

2.

5. 0 ≥ 3x + 3

4. 4x + 1 ≤ 25

x ≤ 7

23 < –2x + 3 –10 > x

Solve each inequality and graph the solutions.

x ≤ 6

–1 ≥ x

SECTION 3.6

Solving Compound Inequalities

5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

California Standards

The inequalities you have seen so far are simple inequalities. When two simple inequalities are combined into one statement by the words AND or OR, the result is called a compound inequality.

Vocabulary

REVIEWING INEQUALITIES SYMBOLS

≥

<

>

≤Less than or equal to

Greater than

Greater than or equal to

Less than

IDENTIFY EACH SYMBOL

1) ≥ 2) <3) =4) ≤5) >6) ≠7) ≈

“is greater than or equal to”

“is less than”

“is equal to”

“ is less than or equal to”

“ is greater than”

“is approximately equal to”

“ is not equal to”

LET’S TRY ONE TOGETHER.Step 1: Make two equations

Step 2: undo addition or subtraction

Step 3: Solve

Step 4: Graph

All real numbers greater than or equal to two AND all real numbers less than six.

4 820 6

2 < x < 6

LET’S TRY ONE TOGETHER. Step 1: Make two equations

Step 2: undo addition or subtraction

Step 3: undo division and multiplication

Step 4: Solve

Step 5: Graph

All real numbers greater than or equal to negative four AND all real numbers less than three.

0 4-2-4 2

22 2 2 -4 < x < 3

Short MethodSolve the compound inequality and graph the solutions.

8 < 3x – 1 ≤ 11

8 < 3x – 1 ≤ 11+1 +1 +1

9 < 3x ≤ 12

3 < x ≤ 4

1. add 1 to each part of the inequality.

2. divide each part of the inequality by 3 to undo the multiplication.

The solution set is {x:3 < x ≤ 4}.

–5–4 –3 –2–1 0 1 2 3 4 5

Graph 3 < x.

Graph x ≤ 4.

Graph the intersection by finding where the two graphs overlap.

3 < x ≤ 4

LET’S TRY ONE TOGETHER. Step 1: It’s already two equations

Step 2: undo addition or subtraction

Step 3: Solve

Step 4: Graph

All real numbers greater than five OR all real numbers less than one.

3 71-1 5 a > 5 OR a < 1

LET’S TRY ONE TOGETHER. Step 1: It’s already two equations

Step 2: undo addition or subtraction

Step 3: Solve

Step 4: Graph

All real numbers greater than five OR all real numbers less than two.

3 71-1 5

a < 2 OR a > 5

MATCH THE COMPOUND INEQUALITY WITH THE CORRECT GRAPH

1. 0 < x + 2 < 5

2. -4 + a > 1 OR -4 + a < -3

3. -3 < x + 2 < 3

4. 2 < x + 2 < 5

5. x + 2 < -6 OR x + 2 > -2

2-2 0

40 2

51 3

0-4 -2

-4-8 -6

NOW YOU TRY…SOLVE AND GRAPH THE COMPOUND INEQUALITY

1. -3 < x + 2 < 7

2. x – 1 < -1 OR x – 5 > -1

3. 2 < x + 2 < 5

4. 11 < 2x + 3 < 21

5. n + 2 < 3 OR n + 3 > 7

5-5 0

40 2

40 2

84 6

51 3

-5 < x < 5

x < 0 OR x > 4

x < 1 OR x > 4

0 < x < 3

4 < x < 9

The compound inequality is x ≤ –8 OR x > 0.

Write the compound inequality shown by the graph.

The compound inequality is –9 < x AND x < –2 (or –9 < x < –2).

Write the compound inequality shown by the graph.

MATCH THE FOLLOWING

1. Inequality

2. Natural Numbers

3. Inverse Operations

4. Like Terms

5. Compound Inequality

A. the set of counting numbers

B. two inequalities that are combined into one statement by the word AND or OR.

C. terms that contain the same variable raised to the same power

D. a mathematical statement that compares two expressions by using one of the following signs: <, >, <, >, or ≠

E. operation that “undo” each other

IN SUMMARY

Today you learned that two inequalities that are combined into one statement by the word AND or OR is called a compound inequality.

If it contains the word AND it is split into two equations and the graph is in between two points.

If it contains the word OR the graphs go in opposite directions from each point.

Lesson Quiz

Solve each compound inequality and graph the solutions.

1. 2 ≤ 2w + 4 ≤ 12

–1 ≤ w ≤ 4

2. 3 + r > −2 OR 3 + r < −7

r > –5 OR r < –10

Write the compound inequality shown by each graph.

4. x < −7 OR x ≥ 0

5.−2 ≤ a < 4