6-4A Solving Compound Inequalities Involving “AND”

Algebra 1 Glencoe McGraw-Hill Linda Stamper

What is a compound sentence in your English class?This weekend I will go shopping and I will go to a movie.

This weekend I will go shopping or I will go to a movie.

A compound inequality consists of two inequalities connected by the word “AND” or the word “OR”.

Today we will work with the type connected by the word “AND”.

Compound inequalities involving “AND” consist of two inequalities. Both inequalities must be satisfied to make the compound inequality true.

If you want to go to the movies with your friends you must finish your homework andand clean your room.

Writing Compound Inequalities with “AND”.Write a compound inequality that represents the

set of all real numbers greater than or equal to 0 and less than 4. Then graph the inequality.

0n and 4nCompound inequalities with “AND” are generally combined in a single inequality with the variable in the middle.

4n0

0 4O

Both inequalities must be satisfied to make the compound inequality true. Therefore the solution is the intersection (overlap) of the two rays. The graph is a line segment! The number of solutions are limited.

•

Write a compound inequality that represents the set of all real numbers greater than or equal to 2 and less than 8. Then graph the inequality.

2n and 8n8n2

2 8O•

Write the two inequalities.

Rewrite as a single inequality with the variable in the middle.Graph.

Note: A compound inequality is usually written in a way to reflect the order of numbers on a number line.

Example 1 Write a compound inequality that represents the set of all real numbers greater than or equal to -12 and less than -5. Then graph the inequality.

Example 2 Write a compound inequality that represents the set of all real numbers greater than –2 and less than or equal to 3. Then graph the inequality.

Example 3 Magic Mountains newest roller coaster ride has weight restrictions of greater than or equal to 50 pounds and less than 250 pounds.Write a compound inequality to describe the weight restriction and then graph the inequality.

Example 4 Write a compound inequality to describe when water is liquid the temperature is greater than 32 degrees F and less than 212 degrees F. Then graph the inequality.

Example 1 Write a compound inequality that represents the set of all real numbers greater than or equal to -12 and less than -5. Then graph the inequality. 12n and 5n

5n12

-12 -5O•

Note: A compound inequality is usually written in a way to reflect the order of numbers on a number line.

In a compound inequality involving AND, both inequalities must be satisfied to make the compound inequality true. Therefore the solution is the intersection (overlap) of the two rays. The graph is a line segment! The number of solutions are limited.

Example 2 Write a compound inequality that represents the set of all real numbers greater than –2 and less than or equal to 3. Then graph the inequality.

2n and 3n

3n2

–2 3O •

You must label the

endpoints!

You must label the

endpoints!

Example 3 Magic Mountains newest roller coaster ride has weight restrictions of greater than or equal to 50 pounds and less than 250 pounds.Write a compound inequality to describe the weight restriction and then graph the inequality.

50w and 250w

25050 w

50 250• O

Example 4 Write a compound inequality to describe when water is liquid the temperature is greater than 32 degrees F and less than 212 degrees F. Then graph the inequality.

Example 4 Write a compound inequality to describe when water is liquid the temperature is greater than 32 degrees F and less than 212 degrees F. Then graph the inequality.

32d and 212d

21232 d

32 212O O

Write an inequality that describes the graph.

8 3

–8 3• O

x

Note: An inequality sign points toward the smaller value – thus both inequality signs point in the same direction - to the left.

1. Identify the endpoints.

2. Name the variable.

3. Write the inequality sign for each endpoint.

Example 5 Write an inequality that describes the graph.

14 2

–14 –2• •x

1. Identify the endpoints.

2. Name the variable.

3. Write the inequality sign for each endpoint.

Solving Compound Inequalities with “AND”.To solve compound inequalities with “and”, isolate

the variable. You must perform the operation on all three expressions. 42x2

2 2x4

–4 2

2 2

• O

Solve. Then graph the solution.

73x25 Example 6

Example 7

Example 8

5 x213

7 1x22

Example 6 Solve – 5 < 2x + 3 < 7. Then graph the solution.

73x25 34x28

–4 2

3 3

2 2 2 2x4

• O

>>

Example 7 Solve –3 < –1 – 2x < 5. Then graph the solution. 5 x213

16 x22

–3 1

1 1

22 23x1

/ /

1x3 O•

Reverse both inequality symbols.

>>

Example 8 Solve –2 < –2x + 1 < 7. Then graph the solution. 7 1x22

16 x23

–3 32

1 1

22 2

3x23

/ /

23

x3

O•

Reverse both inequality symbols.

To solve a compound inequality involving “AND”, isolate the variable (in the center).

To perform any operation on a compound inequality involving “AND”, you must

perform the operation on all three expressions.

The graph of the solutions to a compound inequality involving “AND” is a line

segment.

6-A5 Handout A5 (Study Guide and Intervention Page 27).