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Solution - Absolute value equations

Exact form: x=72
x=\frac{7}{2}
Mixed number form: x=312
x=3\frac{1}{2}
Decimal form: x=3.5
x=3.5

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

|x+7|+|x|=0

Add |x| to both sides of the equation:

|x+7|+|x||x|=|x|

Simplify the arithmetic

|x+7|=|x|

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

|x|=|y||x+7|=|x|
x=+y(x+7)=(x)
x=y(x+7)=(x)
+x=y(x+7)=(x)
x=y(x+7)=(x)

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+7|=|x|
x=+y , +x=y(x+7)=(x)
x=y , x=y(x+7)=(x)

3. Solve the two equations for x

10 additional steps

(-x+7)=x

Subtract from both sides:

(-x+7)-x=x-x

Group like terms:

(-x-x)+7=x-x

Simplify the arithmetic:

2x+7=xx

Simplify the arithmetic:

2x+7=0

Subtract from both sides:

(-2x+7)-7=0-7

Simplify the arithmetic:

2x=07

Simplify the arithmetic:

2x=7

Divide both sides by :

(-2x)-2=-7-2

Cancel out the negatives:

2x2=-7-2

Simplify the fraction:

x=-7-2

Cancel out the negatives:

x=72

4 additional steps

(-x+7)=-x

Add to both sides:

(-x+7)+x=-x+x

Group like terms:

(-x+x)+7=-x+x

Simplify the arithmetic:

7=x+x

Simplify the arithmetic:

7=0

The statement is false:

7=0

The equation is false so it has no solution.

4. List the solutions

x=72
(1 solution(s))

5. Graph

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