1.6 Absolute-Value Equations & Inequalities

• Absolute value of a number is its distance from zero on the number line. Absolute value of a number is never negative.

• If x ≥ 0, then |x| = x

• If x < 0, then |x| = -x

Properties of Absolute ValueFor any real number a and b

|ab| = |a| |b|

a |a|

b |b| b ≠ 0

|-a| = |a|

=

Simplify

1) |4x|=

2) |-4x2|=

3) |x10|=

4) |x9|=

5) 6x3 =

-3x2

The distance between 2 numbers a and b is |a-b| or |b-a|

1) Find the distance between -8 and 1

Answer: 9

2) Find the distance between -6 and -35

Answer: 29

Solving equations with Absolute ValueIf |x| = p (p is positive)then x = -p or x = p

If |x| = 0 then x = 0

If |x| = -p then there is no solution

If |x| = |p| then x = -p or x = p

Examples

• On the black board

Solving inequalties with Absolute Value

If |x| < p (p is positive)then -p < x < p

If |x| > pthen x < -p or x > p

If |x| < -p then there is no solution

Examples

• On the black board