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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 without absolute value bars

Use the rules:
|x|=|y| → x=±y and |x|=|y| → ±x=y
to write all four options of the equation
|4n|=2|n|
without the absolute value bars:

|x|=|y||4n|=2|n|
x=+y(4n)=2(n)
x=−y(4n)=2(−(n))
+x=y(4n)=2(n)
−x=y−(4n)=2(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||4n|=2|n|
x=+y , +x=y(4n)=2(n)
x=−y , −x=y(4n)=2(−(n))

2. Solve the two equations for n

3 additional steps

4n=2n

Subtract from both sides:

(4n)-2n=(2n)-2n

Simplify the arithmetic:

2n=(2n)-2n

Simplify the arithmetic:

2n=0

Divide both sides by the coefficient:

n=0

5 additional steps

4n=2·-n

Group like terms:

4n=(2·-1)n

Multiply the coefficients:

4n=−2n

Add to both sides:

(4n)+2n=(-2n)+2n

Simplify the arithmetic:

6n=(-2n)+2n

Simplify the arithmetic:

6n=0

Divide both sides by the coefficient:

n=0

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
y=|4n|
y=2|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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