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Solution - Absolute value equations

Exact form: x=-32
x=-\frac{3}{2}
Mixed number form: x=-112
x=-1\frac{1}{2}
Decimal form: x=1.5
x=-1.5

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+4|=|3x+5|
without the absolute value bars:

|x|=|y||3x+4|=|3x+5|
x=+y(3x+4)=(3x+5)
x=y(3x+4)=(3x+5)
+x=y(3x+4)=(3x+5)
x=y(3x+4)=(3x+5)

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+4|=|3x+5|
x=+y , +x=y(3x+4)=(3x+5)
x=y , x=y(3x+4)=(3x+5)

2. Solve the two equations for x

5 additional steps

(3x+4)=(3x+5)

Subtract from both sides:

(3x+4)-3x=(3x+5)-3x

Group like terms:

(3x-3x)+4=(3x+5)-3x

Simplify the arithmetic:

4=(3x+5)-3x

Group like terms:

4=(3x-3x)+5

Simplify the arithmetic:

4=5

The statement is false:

4=5

The equation is false so it has no solution.

12 additional steps

(3x+4)=-(3x+5)

Expand the parentheses:

(3x+4)=-3x-5

Add to both sides:

(3x+4)+3x=(-3x-5)+3x

Group like terms:

(3x+3x)+4=(-3x-5)+3x

Simplify the arithmetic:

6x+4=(-3x-5)+3x

Group like terms:

6x+4=(-3x+3x)-5

Simplify the arithmetic:

6x+4=5

Subtract from both sides:

(6x+4)-4=-5-4

Simplify the arithmetic:

6x=54

Simplify the arithmetic:

6x=9

Divide both sides by :

(6x)6=-96

Simplify the fraction:

x=-96

Find the greatest common factor of the numerator and denominator:

x=(-3·3)(2·3)

Factor out and cancel the greatest common factor:

x=-32

3. Graph

Each line represents the function of one side of the equation:
y=|3x+4|
y=|3x+5|
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.