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Solution - Absolute value equations

Exact form: =-13,-1
=-\frac{1}{3} , -1
Decimal form: =0.333,1
=-0.333 , -1

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
|+1|=|3x+2|
without the absolute value bars:

|x|=|y||+1|=|3x+2|
x=+y(+1)=(3x+2)
x=y(+1)=(3x+2)
+x=y(+1)=(3x+2)
x=y(+1)=(3x+2)

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||+1|=|3x+2|
x=+y , +x=y(+1)=(3x+2)
x=y , x=y(+1)=(3x+2)

2. Solve the two equations for

5 additional steps

(1)=(3x+2)

Swap sides:

(3x+2)=(1)

Subtract from both sides:

(3x+2)-2=(1)-2

Simplify the arithmetic:

3x=(1)-2

Simplify the arithmetic:

3x=1

Divide both sides by :

(3x)3=-13

Simplify the fraction:

x=-13

9 additional steps

(1)=-(3x+2)

Expand the parentheses:

(1)=-3x-2

Swap sides:

-3x-2=(1)

Add to both sides:

(-3x-2)+2=(1)+2

Simplify the arithmetic:

-3x=(1)+2

Simplify the arithmetic:

3x=3

Divide both sides by :

(-3x)-3=3-3

Cancel out the negatives:

3x3=3-3

Simplify the fraction:

x=3-3

Move the negative sign from the denominator to the numerator:

x=-33

Simplify the fraction:

x=1

3. List the solutions

=-13,-1
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|+1|
y=|3x+2|
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.