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Solution - Absolute value equations

Exact form: =4,−6
=4 , -6
See steps

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
|+5|=|w+1|
without the absolute value bars:

|x|=|y||+5|=|w+1|
x=+y(+5)=(w+1)
x=−y(+5)=−(w+1)
+x=y(+5)=(w+1)
−x=y−(+5)=(w+1)

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||+5|=|w+1|
x=+y , +x=y(+5)=(w+1)
x=−y , −x=y(+5)=−(w+1)

2. Solve the two equations for

3 additional steps

(5)=(w+1)

Swap sides:

(w+1)=(5)

Subtract from both sides:

(w+1)-1=(5)-1

Simplify the arithmetic:

w=(5)-1

Simplify the arithmetic:

w=4

7 additional steps

(5)=-(w+1)

Expand the parentheses:

(5)=-w-1

Swap sides:

-w-1=(5)

Add to both sides:

(-w-1)+1=(5)+1

Simplify the arithmetic:

-w=(5)+1

Simplify the arithmetic:

−w=6

Multiply both sides by :

-w·-1=6·-1

Remove the one(s):

w=6·-1

Simplify the arithmetic:

w=−6

3. List the solutions

=4,−6
(2 solution(s))

4. Graph

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

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