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Solution - Absolute value equations

Exact form: y=0,0
y=0 , 0
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

|y|+|y|=0

Add −|y| to both sides of the equation:

|y|+|y|−|y|=−|y|

Simplify the arithmetic

|y|=−|y|

2. 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
|y|=−|y|
without the absolute value bars:

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

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

3. Solve the two equations for y

3 additional steps

y=−y

Add to both sides:

y+y=−y+y

Simplify the arithmetic:

2y=−y+y

Simplify the arithmetic:

2y=0

Divide both sides by the coefficient:

y=0

2 additional steps

y=y

Subtract from both sides:

y−y=y−y

Simplify the arithmetic:

0=y−y

Simplify the arithmetic:

0=0

4. List the solutions

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

5. Graph

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