Absolute Value

**Absolute Value**

**Absolute value** is a number's distance from 0.

|6| = 6

|-8| = 8

|0| = 0

When solving equations or inequalities involving absolute value, it may be helpful to translate the expression into English in order to understand what the expression means.

|states thatx| = 5xis exactly 5 units away from zero.

- This expression may be written
x= 5 orx= -5.

lstates thatxl < 5xis less than 5 units away from zero.

- Either
xis greater than -5 or less than +5. This expression may be written -5 <x< 5.

lstates thatxl > 5xis more than 5 units away from zero.

- Either
xis greater than 5 orxis less than -5.- This expression may be written
x> 5 orx< -5.

**EXAMPLE 1**

Solve the equation |3*x* - 5| = 13.

You can re-express the equation as 3*x* - 5 = 13 or 3*x* - 5 = -13.

Solving these equations separately:

3*x* - 5 + 5 = 13 + 5 3*x* - 5 + 5 = -13 + 5

3*x* = 18 3*x* = -8

3*x* / 3 = 18 / 3 3*x* / 3 = -8 / 3

*x* = 6 *x* = -^{8}/_{3}

The solution to the equation is the set ** x = {-^{8}/_{3}, 6}**.

**EXAMPLE 2**

Solve and graph the inequality |2*x* + 3| < 9.

Re-expressing the inequality as -9 < 2*x* + 3 < 9, we can solve this by operating on all three sides:

-9 - 3 < 2*x* + 3 - 3 < 9 - 3

-12 < 2*x* < 6

-12 / 2 < 2*x* / 2 < 6 / 2

-6 < *x* < 3

The solution is the interval **(-6, 3)**, and is graphed on the number line as .

**Remember... **whenever you divide an inequality by a negative number, the symbol(s) change direction.