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Solution - Absolute value equations

Exact form: z=45,43
z=\frac{4}{5} , \frac{4}{3}
Mixed number form: z=45,113
z=\frac{4}{5} , 1\frac{1}{3}
Decimal form: z=0.8,1.333
z=0.8 , 1.333

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
|2z4|=|7z8|
without the absolute value bars:

|x|=|y||2z4|=|7z8|
x=+y(2z4)=(7z8)
x=y(2z4)=(7z8)
+x=y(2z4)=(7z8)
x=y(2z4)=(7z8)

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||2z4|=|7z8|
x=+y , +x=y(2z4)=(7z8)
x=y , x=y(2z4)=(7z8)

2. Solve the two equations for z

11 additional steps

(2z-4)=(7z-8)

Subtract from both sides:

(2z-4)-7z=(7z-8)-7z

Group like terms:

(2z-7z)-4=(7z-8)-7z

Simplify the arithmetic:

-5z-4=(7z-8)-7z

Group like terms:

-5z-4=(7z-7z)-8

Simplify the arithmetic:

5z4=8

Add to both sides:

(-5z-4)+4=-8+4

Simplify the arithmetic:

5z=8+4

Simplify the arithmetic:

5z=4

Divide both sides by :

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

Cancel out the negatives:

5z5=-4-5

Simplify the fraction:

z=-4-5

Cancel out the negatives:

z=45

12 additional steps

(2z-4)=-(7z-8)

Expand the parentheses:

(2z-4)=-7z+8

Add to both sides:

(2z-4)+7z=(-7z+8)+7z

Group like terms:

(2z+7z)-4=(-7z+8)+7z

Simplify the arithmetic:

9z-4=(-7z+8)+7z

Group like terms:

9z-4=(-7z+7z)+8

Simplify the arithmetic:

9z4=8

Add to both sides:

(9z-4)+4=8+4

Simplify the arithmetic:

9z=8+4

Simplify the arithmetic:

9z=12

Divide both sides by :

(9z)9=129

Simplify the fraction:

z=129

Find the greatest common factor of the numerator and denominator:

z=(4·3)(3·3)

Factor out and cancel the greatest common factor:

z=43

3. List the solutions

z=45,43
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2z4|
y=|7z8|
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.