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Solution - Absolute value equations

Exact form: x=-27,-1
x=-\frac{2}{7} , -1
Decimal form: x=0.286,1
x=-0.286 , -1

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
|4x1|=|10x5|
without the absolute value bars:

|x|=|y||4x1|=|10x5|
x=+y(4x1)=(10x5)
x=y(4x1)=(10x5)
+x=y(4x1)=(10x5)
x=y(4x1)=(10x5)

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||4x1|=|10x5|
x=+y , +x=y(4x1)=(10x5)
x=y , x=y(4x1)=(10x5)

2. Solve the two equations for x

11 additional steps

(4x-1)=(-10x-5)

Add to both sides:

(4x-1)+10x=(-10x-5)+10x

Group like terms:

(4x+10x)-1=(-10x-5)+10x

Simplify the arithmetic:

14x-1=(-10x-5)+10x

Group like terms:

14x-1=(-10x+10x)-5

Simplify the arithmetic:

14x1=5

Add to both sides:

(14x-1)+1=-5+1

Simplify the arithmetic:

14x=5+1

Simplify the arithmetic:

14x=4

Divide both sides by :

(14x)14=-414

Simplify the fraction:

x=-414

Find the greatest common factor of the numerator and denominator:

x=(-2·2)(7·2)

Factor out and cancel the greatest common factor:

x=-27

13 additional steps

(4x-1)=-(-10x-5)

Expand the parentheses:

(4x-1)=10x+5

Subtract from both sides:

(4x-1)-10x=(10x+5)-10x

Group like terms:

(4x-10x)-1=(10x+5)-10x

Simplify the arithmetic:

-6x-1=(10x+5)-10x

Group like terms:

-6x-1=(10x-10x)+5

Simplify the arithmetic:

6x1=5

Add to both sides:

(-6x-1)+1=5+1

Simplify the arithmetic:

6x=5+1

Simplify the arithmetic:

6x=6

Divide both sides by :

(-6x)-6=6-6

Cancel out the negatives:

6x6=6-6

Simplify the fraction:

x=6-6

Move the negative sign from the denominator to the numerator:

x=-66

Simplify the fraction:

x=1

3. List the solutions

x=-27,-1
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|4x1|
y=|10x5|
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.