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Solution - Absolute value equations

Exact form: n=0,0
n=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

|n|+|−n|=0

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

|n|+|−n|−|−n|=−|−n|

Simplify the arithmetic

|n|=−|−n|

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
|n|=−|−n|
without the absolute value bars:

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

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

3. Solve the two equations for n

2 additional steps

n=n

Subtract from both sides:

n−n=n−n

Simplify the arithmetic:

0=n−n

Simplify the arithmetic:

0=0

3 additional steps

n=−n

Add to both sides:

n+n=−n+n

Simplify the arithmetic:

2n=−n+n

Simplify the arithmetic:

2n=0

Divide both sides by the coefficient:

n=0

4. List the solutions

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

5. Graph

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