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Solution - Absolute value equations

Exact form: =821,421
=\frac{8}{21} , \frac{4}{21}
Decimal form: =0.381,0.190
=0.381 , 0.190

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
|+2|=3|7x2|
without the absolute value bars:

|x|=|y||+2|=3|7x2|
x=+y(+2)=3(7x2)
x=y(+2)=3((7x2))
+x=y(+2)=3(7x2)
x=y(+2)=3(7x2)

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||+2|=3|7x2|
x=+y , +x=y(+2)=3(7x2)
x=y , x=y(+2)=3((7x2))

2. Solve the two equations for

8 additional steps

(2)=3·(7x-2)

Expand the parentheses:

(2)=3·7x+3·-2

Multiply the coefficients:

(2)=21x+3·-2

Simplify the arithmetic:

(2)=21x-6

Swap sides:

21x-6=(2)

Add to both sides:

(21x-6)+6=(2)+6

Simplify the arithmetic:

21x=(2)+6

Simplify the arithmetic:

21x=8

Divide both sides by :

(21x)21=821

Simplify the fraction:

x=821

11 additional steps

(2)=3·(-(7x-2))

Expand the parentheses:

(2)=3·(-7x+2)

Expand the parentheses:

(2)=3·-7x+3·2

Multiply the coefficients:

(2)=-21x+3·2

Simplify the arithmetic:

(2)=-21x+6

Swap sides:

-21x+6=(2)

Subtract from both sides:

(-21x+6)-6=(2)-6

Simplify the arithmetic:

-21x=(2)-6

Simplify the arithmetic:

21x=4

Divide both sides by :

(-21x)-21=-4-21

Cancel out the negatives:

21x21=-4-21

Simplify the fraction:

x=-4-21

Cancel out the negatives:

x=421

3. List the solutions

=821,421
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|+2|
y=3|7x2|
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.