Page 206 - ABCTE Study Guide_Neat

"Less Than" Absolute-Value Inequalities

In "less than" and "less than or equal to" absolute-value inequalities, set the absolute value part of the
inequality between the positive and negative values of the rest of the inequality. This is better shown in an
example:

Write this either as a combined inequality:

-12 < n + 3 < 12

Or as two separate inequalities:

- 12 < n + 3 AND n + 3 < 12

Solve by getting the variable alone, the same way that you would in an equation.

n > -15 AND n < 9

Written as a combined inequality, the solution would be:

-15 < n < 9

The only numbers that will make this inequality be true are numbers greater than -15 AND less than 9.

Here is one more example to view:

-25 < k -5 < 25

-20 < k < 30

Values of k between -20 and 30 make the inequality true.

"Greater Than" Absolute-Value Inequalities

In "greater than" and "greater than or equal to" absolute-value inequalities, you have to create inequalities
that go in opposite directions. The first inequality will have the absolute value be less than the negative
value, and the second inequality will have the absolute value greater than the positive value.

Let's look at two examples that show how this works.

1)

This inequality means that the value of (n+7) will be greater than or equal to 11 or less than or equal to -
11.