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Solution - Absolute value equations

Exact form: x=3,43
x=3 , \frac{4}{3}
Mixed number form: x=3,113
x=3 , 1\frac{1}{3}
Decimal form: x=3,1.333
x=3 , 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
|x+2|=5|x2|
without the absolute value bars:

|x|=|y||x+2|=5|x2|
x=+y(x+2)=5(x2)
x=y(x+2)=5((x2))
+x=y(x+2)=5(x2)
x=y(x+2)=5(x2)

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||x+2|=5|x2|
x=+y , +x=y(x+2)=5(x2)
x=y , x=y(x+2)=5((x2))

2. Solve the two equations for x

15 additional steps

(x+2)=5·(x-2)

Expand the parentheses:

(x+2)=5x+5·-2

Simplify the arithmetic:

(x+2)=5x-10

Subtract from both sides:

(x+2)-5x=(5x-10)-5x

Group like terms:

(x-5x)+2=(5x-10)-5x

Simplify the arithmetic:

-4x+2=(5x-10)-5x

Group like terms:

-4x+2=(5x-5x)-10

Simplify the arithmetic:

4x+2=10

Subtract from both sides:

(-4x+2)-2=-10-2

Simplify the arithmetic:

4x=102

Simplify the arithmetic:

4x=12

Divide both sides by :

(-4x)-4=-12-4

Cancel out the negatives:

4x4=-12-4

Simplify the fraction:

x=-12-4

Cancel out the negatives:

x=124

Find the greatest common factor of the numerator and denominator:

x=(3·4)(1·4)

Factor out and cancel the greatest common factor:

x=3

16 additional steps

(x+2)=5·(-(x-2))

Expand the parentheses:

(x+2)=5·(-x+2)

(x+2)=5·-x+5·2

Group like terms:

(x+2)=(5·-1)x+5·2

Multiply the coefficients:

(x+2)=-5x+5·2

Simplify the arithmetic:

(x+2)=-5x+10

Add to both sides:

(x+2)+5x=(-5x+10)+5x

Group like terms:

(x+5x)+2=(-5x+10)+5x

Simplify the arithmetic:

6x+2=(-5x+10)+5x

Group like terms:

6x+2=(-5x+5x)+10

Simplify the arithmetic:

6x+2=10

Subtract from both sides:

(6x+2)-2=10-2

Simplify the arithmetic:

6x=102

Simplify the arithmetic:

6x=8

Divide both sides by :

(6x)6=86

Simplify the fraction:

x=86

Find the greatest common factor of the numerator and denominator:

x=(4·2)(3·2)

Factor out and cancel the greatest common factor:

x=43

3. List the solutions

x=3,43
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|x+2|
y=5|x2|
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.