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.

|x| = 5 states that x is exactly 5 units away from zero.

• This expression may be written as x = 5 or x = −5.
• The solution may be expressed with a set:  {−5, 5}.

|x| < 5  states that x is less than 5 units away from zero.

• Either x is 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).

|x| > 5  states that x is more than 5 units away from zero.

• Either x is greater than 5 or x is less than −5.
• This expression may be written x > 5 or x < −5.
• The solution may be expressed with a union of intervals:  (−, −5) ⋃ (5, ).

EXAMPLE 1

Solve the equation |3x − 5| = 13.

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

Solving these equations separately:

3x − 5 + 5 = 13 + 5          3x − 5 + 513 + 5

3x = 18                            3x8

3x/3 = 18/3                     3x/3 = 8/3

x = 6                                x8/3

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

EXAMPLE 2

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

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

9 − 3 < 2x + 3 − 3 < 9 − 3

12 < 2x < 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 − 2x| = 15

3.  |5x + 1| − 3 = 16

4.  |3x − 2| ≤ 7

5.  |6x + 5| > 9

6.  |4x + 3| < −5

1.  {−16, 8}

2.  {−4, 11}

3.  {−4, }

4.  [, 3]

5.  (−, ) ⋃ (, )

6.  ⊘