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Solution - Absolute value equations

Exact form: =-54,54
=-\frac{5}{4} , \frac{5}{4}
Mixed number form: =-114,114
=-1\frac{1}{4} , 1\frac{1}{4}
Decimal form: =−1.25,1.25
=-1.25 , 1.25
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
|−5|=4|z|
without the absolute value bars:

|x|=|y||−5|=4|z|
x=+y(−5)=4(z)
x=−y(−5)=4(−(z))
+x=y(−5)=4(z)
−x=y−(−5)=4(z)

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||−5|=4|z|
x=+y , +x=y(−5)=4(z)
x=−y , −x=y(−5)=4(−(z))

2. Solve the two equations for

2 additional steps

−5=4z

Swap sides:

4z=−5

Divide both sides by :

(4z)4=-54

Simplify the fraction:

z=-54

6 additional steps

-5=4·-z

Group like terms:

-5=(4·-1)z

Multiply the coefficients:

−5=−4z

Swap sides:

−4z=−5

Divide both sides by :

(-4z)-4=-5-4

Cancel out the negatives:

4z4=-5-4

Simplify the fraction:

z=-5-4

Cancel out the negatives:

z=54

3. List the solutions

=-54,54
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|−5|
y=4|z|
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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