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Solution - Absolute value equations

Exact form: =7,1
=7 , 1
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−4|
without the absolute value bars:

|x|=|y||+3|=|z−4|
x=+y(+3)=(z−4)
x=−y(+3)=−(z−4)
+x=y(+3)=(z−4)
−x=y−(+3)=(z−4)

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−4|
x=+y , +x=y(+3)=(z−4)
x=−y , −x=y(+3)=−(z−4)

2. Solve the two equations for

3 additional steps

(3)=(z-4)

Swap sides:

(z-4)=(3)

Add to both sides:

(z-4)+4=(3)+4

Simplify the arithmetic:

z=(3)+4

Simplify the arithmetic:

z=7

7 additional steps

(3)=-(z-4)

Expand the parentheses:

(3)=-z+4

Swap sides:

-z+4=(3)

Subtract from both sides:

(-z+4)-4=(3)-4

Simplify the arithmetic:

-z=(3)-4

Simplify the arithmetic:

−z=−1

Multiply both sides by :

-z·-1=-1·-1

Remove the one(s):

z=-1·-1

Simplify the arithmetic:

z=1

3. List the solutions

=7,1
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|+3|
y=|z−4|
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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