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Solution - Absolute value equations

Exact form: =3,7
=3 , 7
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
|+2|=|−x+5|
without the absolute value bars:

|x|=|y||+2|=|−x+5|
x=+y(+2)=(−x+5)
x=−y(+2)=−(−x+5)
+x=y(+2)=(−x+5)
−x=y−(+2)=(−x+5)

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||+2|=|−x+5|
x=+y , +x=y(+2)=(−x+5)
x=−y , −x=y(+2)=−(−x+5)

2. Solve the two equations for

6 additional steps

(2)=(-x+5)

Swap sides:

(-x+5)=(2)

Subtract from both sides:

(-x+5)-5=(2)-5

Simplify the arithmetic:

-x=(2)-5

Simplify the arithmetic:

−x=−3

Multiply both sides by :

-x·-1=-3·-1

Remove the one(s):

x=-3·-1

Simplify the arithmetic:

x=3

4 additional steps

(2)=-(-x+5)

Expand the parentheses:

(2)=x-5

Swap sides:

x-5=(2)

Add to both sides:

(x-5)+5=(2)+5

Simplify the arithmetic:

x=(2)+5

Simplify the arithmetic:

x=7

3. List the solutions

=3,7
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|+2|
y=|−x+5|
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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