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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
3|x|=3|x|
without the absolute value bars:

|x|=|y|3|x|=3|x|
x=+y3(x)=3(x)
x=−y3(x)=3(−(x))
+x=y3(x)=3(x)
−x=y3(−(x))=3(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|3|x|=3|x|
x=+y , +x=y3(x)=3(x)
x=−y , −x=y3(x)=3(−(x))

2. Solve the two equations for x

2 additional steps

3x=3x

Subtract from both sides:

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

Simplify the arithmetic:

0=(3x)-3x

Simplify the arithmetic:

0=0

5 additional steps

3x=3·-x

Group like terms:

3x=(3·-1)x

Multiply the coefficients:

3x=−3x

Add to both sides:

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

Simplify the arithmetic:

6x=(-3x)+3x

Simplify the arithmetic:

6x=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=3|x|
y=3|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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