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Solution - Absolute value equations

Exact form: =−4,2
=-4 , 2
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
|−3|=|z+1|
without the absolute value bars:

|x|=|y||−3|=|z+1|
x=+y(−3)=(z+1)
x=−y(−3)=−(z+1)
+x=y(−3)=(z+1)
−x=y−(−3)=(z+1)

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||−3|=|z+1|
x=+y , +x=y(−3)=(z+1)
x=−y , −x=y(−3)=−(z+1)

2. Solve the two equations for

3 additional steps

-3=(z+1)

Swap sides:

(z+1)=-3

Subtract from both sides:

(z+1)-1=-3-1

Simplify the arithmetic:

z=−3−1

Simplify the arithmetic:

z=−4

7 additional steps

-3=-(z+1)

Expand the parentheses:

−3=−z−1

Swap sides:

−z−1=−3

Add to both sides:

(-z-1)+1=-3+1

Simplify the arithmetic:

−z=−3+1

Simplify the arithmetic:

−z=−2

Multiply both sides by :

-z·-1=-2·-1

Remove the one(s):

z=-2·-1

Simplify the arithmetic:

z=2

3. List the solutions

=−4,2
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|−3|
y=|z+1|
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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