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 as
x= 5 orx= −5.- The solution may be expressed with a set: {−5, 5}.

states that|x| < 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. The solution may be expressed with an interval: (−5, 5).

|states thatx| > 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.- The solution may be expressed with a union of intervals: (−∞, −5) ⋃ (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

^{3x}/_{3} = ^{18}/_{3} ^{3x}/_{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} < ^{2x}/_{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.

**Practice**

Solve the following equations.

1. |*x* + 4| = 12

2. |7 − 2*x*| = 15

3. |5*x* + 1| − 3 = 16

Solve the following inequalities. Express your answers as intervals.

4. |3*x* − 2| ≤ 7

5. |6*x* + 5| > 9

6. |4*x* + 3| < −5

**Answers**

1. {−16, 8}

2. {−4, 11}

3. {−4, }

4. [, 3]

5. (−∞, ) ⋃ (, ∞)

6. ⊘