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Solution - Absolute value equations

Exact form: a=4,2
a=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
|2a3|=|3a7|
without the absolute value bars:

|x|=|y||2a3|=|3a7|
x=+y(2a3)=(3a7)
x=y(2a3)=(3a7)
+x=y(2a3)=(3a7)
x=y(2a3)=(3a7)

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||2a3|=|3a7|
x=+y , +x=y(2a3)=(3a7)
x=y , x=y(2a3)=(3a7)

2. Solve the two equations for a

10 additional steps

(2a-3)=(3a-7)

Subtract from both sides:

(2a-3)-3a=(3a-7)-3a

Group like terms:

(2a-3a)-3=(3a-7)-3a

Simplify the arithmetic:

-a-3=(3a-7)-3a

Group like terms:

-a-3=(3a-3a)-7

Simplify the arithmetic:

a3=7

Add to both sides:

(-a-3)+3=-7+3

Simplify the arithmetic:

a=7+3

Simplify the arithmetic:

a=4

Multiply both sides by :

-a·-1=-4·-1

Remove the one(s):

a=-4·-1

Simplify the arithmetic:

a=4

12 additional steps

(2a-3)=-(3a-7)

Expand the parentheses:

(2a-3)=-3a+7

Add to both sides:

(2a-3)+3a=(-3a+7)+3a

Group like terms:

(2a+3a)-3=(-3a+7)+3a

Simplify the arithmetic:

5a-3=(-3a+7)+3a

Group like terms:

5a-3=(-3a+3a)+7

Simplify the arithmetic:

5a3=7

Add to both sides:

(5a-3)+3=7+3

Simplify the arithmetic:

5a=7+3

Simplify the arithmetic:

5a=10

Divide both sides by :

(5a)5=105

Simplify the fraction:

a=105

Find the greatest common factor of the numerator and denominator:

a=(2·5)(1·5)

Factor out and cancel the greatest common factor:

a=2

3. List the solutions

a=4,2
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2a3|
y=|3a7|
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.