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.
|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 + 5 = −13 + 5
3x = 18 3x = −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 |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
Solve the following inequalities. Express your answers as intervals.
4. |3x − 2| ≤ 7
5. |6x + 5| > 9
6. |4x + 3| < −5
Answers
1. {−16, 8}
2. {−4, 11}
3. {−4, }
4. [, 3]
5. (−∞, ) ⋃ (, ∞)
6. ⊘