Solving Compound Inequalities Section 4.3 Solving Compound Inequalities

4.3 Lecture Guide: Solving Compound Inequalities Objective 1: 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. They 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 _____________________ inequality is an inequality that is true for some but not all values of the variable. An _____________________ inequality 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.

1. Identify each inequality as a conditional inequality, an unconditional inequality, or a contradiction. (a) (b) (c)

Solve each inequality. Identify each contradiction or unconditional inequality. 2. 3.

Identify each inequality as a conditional inequality, a contradiction or an unconditional inequality and solve. Type: Solution: 4.

Identify each inequality as a conditional inequality, a contradiction or an unconditional inequality and solve. Type: Solution: 5.

Identify each inequality as a conditional inequality, a contradiction or an unconditional inequality and solve. Type: Solution: 6.

Objective 2: Write the intersection or union of two intervals. Intersection of Two Sets Algebraic Notation Verbally The intersection of A and B is the set that contains the elements in __________ A and B. Numerical Example Graphical Example

Union of Two Sets Algebraic Notation Verbally The __________ of A and B is the set that contains the elements in either A or B or both. Numerical Example Graphical Example

Write each inequality as two separate inequalities using the word “and” to connect the inequalities. 7. 8.

Write each inequality as a single compound inequality. 9. and 10. and

Represent each union of intervals by two separate inequalities, using the word or to connect the inequalities. 11. 12.

Consider the sets and . 13. Determine . 14. Determine .

15. (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 and their union . 16. ; = ____________ = ____________

Graph each pair of intervals on the same number line and then give both their intersection and their union . 17. ; = ____________ = ____________

Graph each pair of intervals on the same number line and then give both their intersection and their union . 18. ; = ____________ = ____________

Graph each pair of intervals on the same number line and then give both their intersection and their union . 19. ; = ____________ = ____________

20. Complete the following table. Compound Inequality Verbal Description Graph Interval Notation and or

Objective 3: Solve compound inequalities involving intersection and union. Solve each compound inequality. Give the solution in interval notation. 21. 22.

Solve each compound inequality. Give the solution in interval notation. 23.

Solve each compound inequality. Give the solution in interval notation. 24.

Solve each compound inequality. Give the solution in interval notation. 25. or

Solve each compound inequality. Give the solution in interval notation. 26. and

Solve each compound inequality. Give the solution in interval notation. 27. or

Solve each compound inequality. Give the solution in interval notation. 28. and

29. Use the graph below to determine the solution of

30. A golfer must hit a shot more than 170 yd to clear a lake in front of a green. The depth of the green is only 20 yd. Write a compound inequality that expresses the length of the tee shot that a golfer must make to clear the water and to land safely on the green of this golf hole. 170 yards 20 yards

31. 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 x cm