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Solution - Absolute value equations

Exact form: x=194,-72
x=\frac{19}{4} , -\frac{7}{2}
Mixed number form: x=434,-312
x=4\frac{3}{4} , -3\frac{1}{2}
Decimal form: x=4.75,3.5
x=4.75 , -3.5

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
|5x+1|=|x+20|
without the absolute value bars:

|x|=|y||5x+1|=|x+20|
x=+y(5x+1)=(x+20)
x=y(5x+1)=(x+20)
+x=y(5x+1)=(x+20)
x=y(5x+1)=(x+20)

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||5x+1|=|x+20|
x=+y , +x=y(5x+1)=(x+20)
x=y , x=y(5x+1)=(x+20)

2. Solve the two equations for x

9 additional steps

(5x+1)=(x+20)

Subtract from both sides:

(5x+1)-x=(x+20)-x

Group like terms:

(5x-x)+1=(x+20)-x

Simplify the arithmetic:

4x+1=(x+20)-x

Group like terms:

4x+1=(x-x)+20

Simplify the arithmetic:

4x+1=20

Subtract from both sides:

(4x+1)-1=20-1

Simplify the arithmetic:

4x=201

Simplify the arithmetic:

4x=19

Divide both sides by :

(4x)4=194

Simplify the fraction:

x=194

12 additional steps

(5x+1)=-(x+20)

Expand the parentheses:

(5x+1)=-x-20

Add to both sides:

(5x+1)+x=(-x-20)+x

Group like terms:

(5x+x)+1=(-x-20)+x

Simplify the arithmetic:

6x+1=(-x-20)+x

Group like terms:

6x+1=(-x+x)-20

Simplify the arithmetic:

6x+1=20

Subtract from both sides:

(6x+1)-1=-20-1

Simplify the arithmetic:

6x=201

Simplify the arithmetic:

6x=21

Divide both sides by :

(6x)6=-216

Simplify the fraction:

x=-216

Find the greatest common factor of the numerator and denominator:

x=(-7·3)(2·3)

Factor out and cancel the greatest common factor:

x=-72

3. List the solutions

x=194,-72
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|5x+1|
y=|x+20|
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.