N−5 −2 - California State University, Northridge

3.6 Inequalities, Compound Inequalities, and Problem Solving I. Inequalities Example 1. Solve each inequality 4. 4n - 3 5n + 6

20. - 2(y + 4) > 18

30. 4(n - 5) - 2(n -1) < 13

42. 0.08x + 0.09(2x) 130

46.

x

- 8

6

-

x

+ 7

2

>

-1

II. Compound Inequalities Compound Inequalities--two inequalities joined by the word "and" or the word "or".

Example of compound inequalities: x - 2 < 9 and 3x + 1 < 7 ; 5x - 3 11 or 9x + 8 > -5 Fact: 1. Compound inequality using the word "and" is true if and only if both inequalities are true.

2. Compound inequality using the word "or" is true if and only if one or both inequalities are true.

A. Compound Inequalities Involving "and" (intersection), ( ) Solution Set for intersection--is the intersection of the two solution sets.

Example 2. Given set A = {1, 2, 5, 7} and set B = {3, 5, 7, 9, 11}. Find the solution set of A or B.

Graph of the solution set for "and"--Graph each inequality on a number line and take the intersection of these solution sets.

Example 3. Graph the following compound inequality. 1. x 2 and x 5

2. x > 1 and x < 4

3. x < -3 and x > 6

B. Compound Inequalities Involving "or" (union), ( ) Solution Set for union--is all the elements of both sets.

Example 4. Given set A = {0,1, 2, 3, 4} and set B = {2, 4, 6}. Find the solution set of A and B.

Graph of the solution set for "or"--Graph each inequality on a number line and take the union of these solution sets.

Example 5. Graph the following compound inequality. 1. x 5 or x > -2

2. x < 2 or x < 4

3. x -2 or x 1

C. Word Problems Involving Inequality Key words:

Is less than

is greater than

Is less than or equal to Maximum At most Or less

is greater than or equal to Minimum At least Or more

Example 6. Solve each problem by setting up and solving an appropriate inequality.

68. Fourteen increased by twice a number is less than or equal to three times the number. Find the numbers that satisfy this relationship.

69. Suppose that the perimeter of a rectangle is to be no greater than 70 inches, and the length of the rectangle must be 20 inches. Find the largest possible value for the width of the rectangle.

72. Mike has scores of 87, 81 and 74 on his first three algebra tests. What score must he get on the fourth test to have an average of 85 or higher for the four tests?

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