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Solution - Absolute value equations

Exact form: =−36,0
=-36 , 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
|−18|=|x+18|
without the absolute value bars:

|x|=|y||−18|=|x+18|
x=+y(−18)=(x+18)
x=−y(−18)=−(x+18)
+x=y(−18)=(x+18)
−x=y−(−18)=(x+18)

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||−18|=|x+18|
x=+y , +x=y(−18)=(x+18)
x=−y , −x=y(−18)=−(x+18)

2. Solve the two equations for

3 additional steps

-18=(x+18)

Swap sides:

(x+18)=-18

Subtract from both sides:

(x+18)-18=-18-18

Simplify the arithmetic:

x=−18−18

Simplify the arithmetic:

x=−36

7 additional steps

-18=-(x+18)

Expand the parentheses:

−18=−x−18

Swap sides:

−x−18=−18

Add to both sides:

(-x-18)+18=-18+18

Simplify the arithmetic:

−x=−18+18

Simplify the arithmetic:

−x=0

Multiply both sides by :

-x·-1=0·-1

Remove the one(s):

x=0·-1

Multiply by zero:

x=0

3. List the solutions

=−36,0
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|−18|
y=|x+18|
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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