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Solution - Absolute value equations

Exact form: v=0,0
v=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
|4v|=|4v|
without the absolute value bars:

|x|=|y||4v|=|4v|
x=+y(4v)=(4v)
x=−y(4v)=−(4v)
+x=y(4v)=(4v)
−x=y−(4v)=(4v)

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||4v|=|4v|
x=+y , +x=y(4v)=(4v)
x=−y , −x=y(4v)=−(4v)

2. Solve the two equations for v

2 additional steps

4v=4v

Subtract from both sides:

(4v)-4v=(4v)-4v

Simplify the arithmetic:

0=(4v)-4v

Simplify the arithmetic:

0=0

6 additional steps

4v=−4v

Divide both sides by :

(4v)4=(-4v)4

Simplify the fraction:

v=(-4v)4

Simplify the fraction:

v=−v

Add to both sides:

v+v=−v+v

Simplify the arithmetic:

2v=−v+v

Simplify the arithmetic:

2v=0

Divide both sides by the coefficient:

v=0

3. List the solutions

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

4. Graph

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