1. Algebra 1
Section 4.6

2. Absolute Value
One way to think of the absolute value of a number is to think of its graph’s distance from the origin.
However, we need a definition for more complicated expressions.

3. Absolute Value
The absolute value of a number, x, is defined as follows:
x if x ≥ 0
-x if x < 0
|x| =

4. Absolute Value Consider |x| = 4.
Since we do not know whether x is positive or negative, there are two possible solutions:
x = 4; x = -4
Or we can write: x = ±4

5. Absolute Value
|x| = 2.5
The two solutions are
x = 2.5 or x = -2.5

6. Absolute Value An absolute value may not have two solutions.
|x| = 0 has just one solution.
|x| = -4 has no solutions, since no number has a negative absolute value.

7. Solving Absolute Value Equations
Simplify within the absolute value symbols.
Isolate the absolute value on one side of the equation.
Use the definition of absolute value to write two equations.

8. Solving Absolute Value Equations
Solve each equation separately.
Check your solutions.

9. Example 1
|2x + 5| = 11
Since the expression within the absolute value symbols must equal 11 or -11:
2x + 5 = 11 or 2x + 5 = -11
x = 3 or x = -8

10. Example 2 5 + |2y + 4 – 5y| = 21 5 + |-3y + 4| = 21 |-3y + 4| = 16
-3y + 4 = 16 or -3y + 4 = -16
y = 3 20
y = -4 or

11. Another Example |x + 5| = -3
This absolute value equation has no solution (Ø). Why?
Because an absolute value of any expression cannot equal a negative number.

12. More Examples |a – b| and |b – a| are equal.
|x – 5| = 3 has two solutions: x = 2 and x = 8.
Both solutions are a distance of 3 units from 5 on a number line.

13. More Examples
5 is the coordinate of the midpoint of the segment having endpoints at 2 and 8.

14. Example 3 4 units from 6 |x – 6| = 4 x – 6 = 4 or x – 6 = -4

15. Example 3 2 units from -7 |x - (-7)| = 2 |x + 7| = 2
x + 7 = 2 or x + 7 = -2
x = -5 or x = -9

16. Homework: pp