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Solution - Absolute value equations

Exact form: x=7,-215
x=7 , -\frac{21}{5}
Mixed number form: x=7,-415
x=7 , -4\frac{1}{5}
Decimal form: x=7,4.2
x=7 , -4.2

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
|3x+7|=|2x+14|
without the absolute value bars:

|x|=|y||3x+7|=|2x+14|
x=+y(3x+7)=(2x+14)
x=y(3x+7)=(2x+14)
+x=y(3x+7)=(2x+14)
x=y(3x+7)=(2x+14)

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||3x+7|=|2x+14|
x=+y , +x=y(3x+7)=(2x+14)
x=y , x=y(3x+7)=(2x+14)

2. Solve the two equations for x

7 additional steps

(3x+7)=(2x+14)

Subtract from both sides:

(3x+7)-2x=(2x+14)-2x

Group like terms:

(3x-2x)+7=(2x+14)-2x

Simplify the arithmetic:

x+7=(2x+14)-2x

Group like terms:

x+7=(2x-2x)+14

Simplify the arithmetic:

x+7=14

Subtract from both sides:

(x+7)-7=14-7

Simplify the arithmetic:

x=147

Simplify the arithmetic:

x=7

10 additional steps

(3x+7)=-(2x+14)

Expand the parentheses:

(3x+7)=-2x-14

Add to both sides:

(3x+7)+2x=(-2x-14)+2x

Group like terms:

(3x+2x)+7=(-2x-14)+2x

Simplify the arithmetic:

5x+7=(-2x-14)+2x

Group like terms:

5x+7=(-2x+2x)-14

Simplify the arithmetic:

5x+7=14

Subtract from both sides:

(5x+7)-7=-14-7

Simplify the arithmetic:

5x=147

Simplify the arithmetic:

5x=21

Divide both sides by :

(5x)5=-215

Simplify the fraction:

x=-215

3. List the solutions

x=7,-215
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|3x+7|
y=|2x+14|
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.