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Solution - Absolute value equations

Exact form: x=-95,3
x=-\frac{9}{5} , 3
Mixed number form: x=-145,3
x=-1\frac{4}{5} , 3
Decimal form: x=1.8,3
x=-1.8 , 3

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|x-1|=|13x-5|
without the absolute value bars:

|x|=|y|2|x-1|=|13x-5|
x=+y2(x-1)=(13x-5)
x=-y2(x-1)=-(13x-5)
+x=y2(x-1)=(13x-5)
-x=y2(-(x-1))=(13x-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|2|x-1|=|13x-5|
x=+y , +x=y2(x-1)=(13x-5)
x=-y , -x=y2(x-1)=-(13x-5)

2. Solve the two equations for x

21 additional steps

2·(x-1)=(13x-5)

Expand the parentheses:

2x+2·-1=(13x-5)

Simplify the arithmetic:

2x-2=(13x-5)

Subtract from both sides:

(2x-2)-13·x=(13x-5)-13x

Group like terms:

(2x+-13·x)-2=(13·x-5)-13x

Group the coefficients:

(2+-13)x-2=(13·x-5)-13x

Convert the integer into a fraction:

(63+-13)x-2=(13·x-5)-13x

Combine the fractions:

(6-1)3·x-2=(13·x-5)-13x

Combine the numerators:

53·x-2=(13·x-5)-13x

Group like terms:

53·x-2=(13·x+-13x)-5

Combine the fractions:

53·x-2=(1-1)3x-5

Combine the numerators:

53·x-2=03x-5

Reduce the zero numerator:

53x-2=0x-5

Simplify the arithmetic:

53x-2=-5

Add to both sides:

(53x-2)+2=-5+2

Simplify the arithmetic:

53x=-5+2

Simplify the arithmetic:

53x=-3

Multiply both sides by inverse fraction :

(53x)·35=-3·35

Group like terms:

(53·35)x=-3·35

Multiply the coefficients:

(5·3)(3·5)x=-3·35

Simplify the fraction:

x=-3·35

Multiply the fraction(s):

x=(-3·3)5

Simplify the arithmetic:

x=-95

22 additional steps

2·(x-1)=-(13x-5)

Expand the parentheses:

2x+2·-1=-(13x-5)

Simplify the arithmetic:

2x-2=-(13x-5)

Expand the parentheses:

2x-2=-13x+5

Add to both sides:

(2x-2)+13·x=(-13x+5)+13x

Group like terms:

(2x+13·x)-2=(-13·x+5)+13x

Group the coefficients:

(2+13)x-2=(-13·x+5)+13x

Convert the integer into a fraction:

(63+13)x-2=(-13·x+5)+13x

Combine the fractions:

(6+1)3·x-2=(-13·x+5)+13x

Combine the numerators:

73·x-2=(-13·x+5)+13x

Group like terms:

73·x-2=(-13·x+13x)+5

Combine the fractions:

73·x-2=(-1+1)3x+5

Combine the numerators:

73·x-2=03x+5

Reduce the zero numerator:

73x-2=0x+5

Simplify the arithmetic:

73x-2=5

Add to both sides:

(73x-2)+2=5+2

Simplify the arithmetic:

73x=5+2

Simplify the arithmetic:

73x=7

Multiply both sides by inverse fraction :

(73x)·37=7·37

Group like terms:

(73·37)x=7·37

Multiply the coefficients:

(7·3)(3·7)x=7·37

Simplify the fraction:

x=7·37

Multiply the fraction(s):

x=(7·3)7

Simplify the arithmetic:

x=3

3. List the solutions

x=-95,3
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=2|x-1|
y=|13x-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.