Solving Absolute

Value Inequalities

Solving Absolute Value Inequalities

1. ax+b < c, where c > 0

Becomes an “and” problem

Changes to: –c < ax+b < c

2. ax+b > c, where c > 0

Becomes an “or” problem

Changes to: ax+b > c or ax+b < -c

To SOLVE an absolute value inequality:● Step 1: ISOLATE the absolute value expression.● Step 2: Rewrite the inequality as an intersection or a union.

● If you have a you are working with an intersection or an ‘and’ statement.

Remember: “Less thand”

● If you have a you are working with a union or an ‘or’ statement.

Remember: “Greator” When writing the 2nd part of the inequality, you must negate one

value and reverse the direction of the inequality sign.

● Step 3: Solve the compound inequality.

or

or

Example 1:

● |2x + 1| > 7● 2x + 1 < - 7 or 2x + 1 > 7

2x <-8 or 2x > 6

● x < -4 or x > 3

This is an ‘or’ statement. (Greator). Rewrite as a union.

Reverse the inequality sign and negate the right side value of one of the inequalities.

Solve each inequality.

Graph the solution.

3-4

Example 2:

● |x -5|< 3

● - 3< x -5< 3

● 2 < x < 8

This is an ‘and’ statement. (Less thand). Rewrite as an intersection.

The negated value moves to the left and use the same inequality sign for both parts of the inequality (either < or ≤).

Solve each inequality.

Graph the solution.

8 2

Solve & graph.

Becomes an “and” problem

2194 x

219421 x30412 x

2

153 x

-3 7 8

Solve & graph

Isolate the absolute value expression first.

Becomes an “or” problem

11323 x

823 x

823or 823 xx63or 103 xx

2or 3

10 xx

-2 3 4

Solve.

● 1) |y – 3| – 5 > - 4 ● y < 2 or y > 4

● 2) -3|p + 2| +7 ≥ -11● -8 ≤ p ≤ 4

Writing Absolute

Value Inequalities

Write a range as an absolute value.● Calculate the mean of the extremes (range).

● Find the tolerance (subtract the mean from the upper extreme).

● Write as an absolute value inequality.

actual mean tolerance

Example: A manufacturer of sleeping bags suggests

that one model is best suited for temperatures between 30°F and 60°F, inclusive. Write an absolute value inequality for this temperature range.

Let t be the actual temperature.•Calculate the mean:

•Find the tolerance:

•Write as an absolute value inequality:

30 6045

2

60 45 15

45 15t