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Solution - Absolute value equations

Exact form: x=−3,1
x=-3 , 1
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
|x|=|y| → x=±y and |x|=|y| → ±x=y
to write all four options of the equation
|2x|=|x−3|
without the absolute value bars:

|x|=|y||2x|=|x−3|
x=+y(2x)=(x−3)
x=−y(2x)=−(x−3)
+x=y(2x)=(x−3)
−x=y−(2x)=(x−3)

When simplified, equations x=+y and +x=y are the same and equations x=−y and −x=y are the same, so we end up with only 2 equations:

|x|=|y||2x|=|x−3|
x=+y , +x=y(2x)=(x−3)
x=−y , −x=y(2x)=−(x−3)

2. Solve the two equations for x

3 additional steps

2x=(x-3)

Subtract from both sides:

(2x)-x=(x-3)-x

Simplify the arithmetic:

x=(x-3)-x

Group like terms:

x=(x-x)-3

Simplify the arithmetic:

x=−3

7 additional steps

2x=-(x-3)

Expand the parentheses:

2x=−x+3

Add to both sides:

(2x)+x=(-x+3)+x

Simplify the arithmetic:

3x=(-x+3)+x

Group like terms:

3x=(-x+x)+3

Simplify the arithmetic:

3x=3

Divide both sides by :

(3x)3=33

Simplify the fraction:

x=33

Simplify the fraction:

x=1

3. List the solutions

x=−3,1
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2x|
y=|x−3|
The equation is true where the two lines cross.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

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