Absolute value

Absolute Value Equations

The interpretation of absolute value as distance on a number line provides a straightforward approach to solving a variety of equations involving absolute value.

The following general property should seem reasonable from the distance interpretation of absolute value:

|ax+b|=k is equivalent to ax+b=-k or ax+b=k, where k is a positive number.

Solve |x|=5

Solution:
Think in terms of distance between the number and zero, and you will see that x must be 5 or -5. That is, the equation |x|=5 is equivalent to

x= -5 or x=5

The solution set is {-5,5}.

Solve |x+2|=5

Solution:
The number, x+2, must be 5 or -5. Thus |x+2|=5 is equivalent to

x+2=5 or x+2=-5

Solving each equation of the disjunction yields

x+2=-5
x=-7

or
or

x+2=5
x=3

The solution set is: {-7,3}.

Solve 3|x-4|=12

Solution:
To use the definition of absolute value to solve this equation, we must isolate the absolute value on the left side of the equal sign. To do so, we divide by 3 both sides of the equation.

3|x-4|=12 $\;\Rightarrow$ |x-4|=4 $\;\Rightarrow$ x-4=-4 or x-4=4

Solving each equation of the disjunction yields

x-4=-4
x=0

or
or

x-4=4
x=8

The solution set is: {0,8}.