Section 4.3

Solving Compound Inequalities

4.3 Lecture Guide: Solving Compound Inequalities

Objective: Identify an inequality that is a contradiction or an unconditional inequality.

The algebraic process for solving the inequalities we have examined in the first two sections of this chapter has left a variable term on one side of the inequality. These have all been conditional inequalities. Sometimes the algebraic process for solving an inequality will result in the variable being completely removed from the inequality, which means the inequality is a contradiction or an unconditional inequality.A _____________________ _____________________ is an inequality that is only true for certain values of the variable.

An _____________________ _____________________ is an inequality that is true for all values of the variable.

A __________________ is an inequality that is not true for any value of the variable.

Solve each inequality. Identify each contradiction or unconditional inequality.

1. 3 2 4 2x x x

Solve each inequality. Identify each contradiction or unconditional inequality.

2. 5 3 2 3 3x x x

Solve each inequality. Identify each contradiction or unconditional inequality.

3. 4 2 3 7 12x x x

Solve each inequality. Identify each contradiction or unconditional inequality.

4. 3 4 2 5 1x x x

5. 5 2 7 2 4x x x

Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1.

Solution: ____________

Type: __________________

10, 10, 1 by 10, 10, 1 1y

2y

6.

Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1.

Solution: ____________

Type: __________________

10, 10, 1 by 10, 10, 1

1y

2y

2 4 5 6 1x x x x

7.

Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1.

Solution: ____________

Type: __________________

10, 10, 1 by 10, 10, 1

1y

2y

2 3 5 9 6x x x

8.

Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1.

Solution: ____________

Type: __________________

10, 10, 1 by 10, 10, 1

1y

2y

3 6 5 3 15 15x x x

Objective: Solve compound inequalities involving intersection and union.

Intersection of Two Sets

Algebraic Notation

A B

Verbally

The intersection of A and B is the set that contains the elements in both A and B.

Numerical Example

3,4 0,6 0,4

Graphical Example

Algebraic Notation

Verbally

Numerical Example

Graphical Example

Union of Two Sets

A B

The union of A and B is the set that contains the elements in either A or B or both.

3,4 0,6 3,6

9. Complete the following table.

Compound Inequality

Verbal Description

Graph Interval Notation

and

and

or

or

3x 7x

2x 0x

1 5x

4 7x

2x 5x

1x 2x

10. (a) Using the word ____________ between two inequalities indicates the intersection of two sets. In some cases, an intersection can be written in a combined form that looks like one expression sandwiched between two other expressions.

(b) Using the word ____________ between two inequalities indicates the union of two sets.

Graph each pair of intervals on the same number line and then give both their intersection

11. (3,6];A [ 1,4]B

A B = ____________

A B = ____________

A B and their union A B .

12.

A B = ____________

A B = ____________

( 3,9);A [2,17)B

Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B .

13.

A B = ____________

A B = ____________

( ,6];A [1, )B

Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B .

14.

A B = ____________

A B = ____________

( ,2);A (5, )B

Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B .

15.

Write each inequality as two separate inequalities using the word “and” to connect the inequalities.

0 2x

16.

Write each inequality as two separate inequalities using the word “and” to connect the inequalities.

13 3x

Write each inequality expression as a single compound

inequality.

17. 2x and 3x

Write each inequality expression as a single compound

inequality.

18. 10x and 8x

Solve each compound inequality. Give the solution in interval notation.

19. 12 6 24x

Solve each compound inequality. Give the solution in interval notation.

20. 1 2 1 3x

Solve each compound inequality. Give the solution in interval notation.

21. 1 3 42

x

Solve each compound inequality. Give the solution in interval notation.

22. 5 2 6 1 5 4x x x

Solve each compound inequality. Give the solution in interval notation.

23. 3 2 1x x or 4 1 3 2x x

Solve each compound inequality. Give the solution in interval notation.

24. 3 2 6x x and 3 2 12x x

Solve each compound inequality. Give the solution in interval notation.

25. 2 3 5 9x x or 12 3

x x

Solve each compound inequality. Give the solution in interval notation.

26. 4 2 7 16x x 5 1 3 9x x and

27. Use the graph below to determine the solution of 2

6 1 53 3 3x x

x

-8

8

-8 8

y

x

2

21

3y x

1 63x

y

3 53x

y

Solution: ___________________

28. The perimeter of the parallelogram shown must be at least 20 cm and no more than 48 cm. If the given length must be 5 cm, determine the possible lengths for x, the unknown dimension.

5 cm 5 cm

x cm

x cm