# Solving absolute value inequalities

### Solving absolute value inequalities

#### Lessons

$\bullet$ absolute value inequalities: $| \heartsuit |$ < $a$,
solution:$-a$ < $\heartsuit$ < $a$
$\bullet$ absolute value inequalities: $| \heartsuit |$ > $a$,
solution:$\heartsuit$ < $-a$ $\;or\;$ $\heartsuit$ > $a$
• Introduction
$\bullet$ definition of absolute value: $| \heartsuit | =$ distance of $\heartsuit$ from zero
$\bullet$ absolute value inequalities: $| \heartsuit |$ < $a$,
solution:$-a$ < $\heartsuit$ < $a$
$\bullet$ absolute value inequalities: $| \heartsuit |$ > $a$,
solution:$\heartsuit$ < $-a$ $\;or\;$ $\heartsuit$ > $a$

• 1.
Solving Basic Absolute Value Inequalities
Solve:
a)
$|x|$ < $4$
$|x| \leq$ $4$

b)
$|x|$ > $4$
$|x| \geq$ $4$

• 2.
Solving Absolute Value Inequalities Involving "less than"
Solve: $|2x-1|$ < $3$

• 3.
Solving Absolute Value Inequalities Involving "greater than"
Solve:
a)
$|4x-5|$ > $7$

b)
$|x|-5 \geq -1$

• 4.
Multiplying/Dividing an Inequality by a Negative Number
Solve:
a)
$|3-2x| \leq 11$

b)
$|-\frac{x}{6}+\frac{5}{3}|$ > $2$

• 5.
Given a Pair of Inequalities, Determine the Corresponding Absolute Value Inequality
Determine the absolute value inequality statement that corresponds to each inequality:
a)
$-1$ < $x$ < $5$

b)
$x \leq-10$ $\;or\;$ $x \geq 2$

• 6.
Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"
Solve:
a)
$|x+3|$ < $-5$

b)
$|x-4|$ > $-1$