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Solution - Absolute value equations

Exact form: x=0,0
x=0 , 0
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
2|x|=|x|
without the absolute value bars:

|x|=|y|2|x|=|x|
x=+y2(x)=(x)
x=−y2(x)=−(x)
+x=y2(x)=(x)
−x=y2(−(x))=(x)

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|2|x|=|x|
x=+y , +x=y2(x)=(x)
x=−y , −x=y2(x)=−(x)

2. Solve the two equations for x

2 additional steps

2x=x

Subtract from both sides:

(2x)-x=x-x

Simplify the arithmetic:

x=x−x

Simplify the arithmetic:

x=0

3 additional steps

2x=−x

Add to both sides:

(2x)+x=-x+x

Simplify the arithmetic:

3x=−x+x

Simplify the arithmetic:

3x=0

Divide both sides by the coefficient:

x=0

3. List the solutions

x=0,0
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=2|x|
y=|x|
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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