6.3 COMPOUND INEQUALITIES

Objectives: Graph the solution sets of compound inequalities. Solve compound inequalities.

Standards Addressed: 2.8.8.C: Create and interpret inequalities that model problem situations. 2.8.8.E: Select and use a strategy to solve an inequality and check the solution.

Ex. 1

b. Alan bought an aquarium and some fish at the pet store. He was told that his fish required water with at least 6 but no more than 10 parts per million of dissolved oxygen. Write and graph a compound inequality to describe this situation.

F > 6 and F < 10 6 < F < 10

Notice that in Example 1a. that c > 1 AND c < 3 and 1b. x > 6 AND x < 10. A compound inequality involving AND has a solution region that represents an intersection, or overlapping, of the solution regions for the separate parts of the inequality. These two examples of compound inequalities are called a conjunction.

Ex. 2 a. In a certain restricted air space, airplanes must fly below 10,000 feet or above 15,000 feet. Write and graph a compound inequality to describe this situation.

A < 10,000 OR A > 15,000

b. On certain highways, it is illegal for a car to drive less than 45 or more than 65 miles per hour. Write and graph a compound inequality to describe the speeds that are prohibited.

C < 45 or C > 65

Example 2 a and b illustrates the other type of compound inequality, a disjunction. A compound inequality involving OR has solution regions that are the union, or the total, of the solution regions of the separate parts of the inequality.

Ex. 3

Ex. 4 a. x > 4 AND x < 3b. -3 < y < 6

c. q < 5 OR q > 16

A. No solution !!

B. Y > -3 AND Y < 6

Q < 5 or Q > 16

Ex. 5a.

Ex. 5 b. Solve and graph -6 < 2x + 4 < 10. -6 < 2x + 4 -10 < 2x -5 < x

AND

2x + 4 < 10 2x < 6 x < 3

-5 < x < 3

Ex. 6a.

b. Solve and graph 5x < 15 OR 3x – 7 > 23. 5x < 15 x < 3

OR

3x – 7 > 23 3x > 30 x > 10

X < 3 OR x > 10

Ex. 7a. Solve and graph 2x + 1 < 13 OR x – 5 > -5.

2x + 1 < 13 2x < 12 x < 6

OR

x – 5 > -5

ALL REALS!!!

Ex. 7 b. Solve and graph 7x + 8 < 43 OR x – 16 > -13.

7x + 8 < 43 7x < 35 x < 5

OR

x – 16 > -13 x > 3

ALL REALS!!

AND: intersection or no solution

OR: union or All Real #s