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Solution - Absolute value equations

Exact form:

x = \frac{7}{3}, \frac{13}{5}

x=\frac{7}{3} , \frac{13}{5}

Mixed number form:

x = 2 \frac{1}{3}, 2 \frac{3}{5}

x=2\frac{1}{3} , 2\frac{3}{5}

Decimal form:

x = 2.333, 2.6

x=2.333 , 2.6

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$\frac{1}{2} | x - 3 | = | 2 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\frac{1}{2} \| x - 3 \| = \| 2 x - 5 \|$
$x = + y$	$\frac{1}{2} (x - 3) = (2 x - 5)$
$x = - y$	$\frac{1}{2} (x - 3) = - (2 x - 5)$
$+ x = y$	$\frac{1}{2} (x - 3) = (2 x - 5)$
$- x = y$	$\frac{1}{2} (- (x - 3)) = (2 x - 5)$

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:

$\| x \| = \| y \|$	$\frac{1}{2} \| x - 3 \| = \| 2 x - 5 \|$
$x = + y$ , $+ x = y$	$\frac{1}{2} (x - 3) = (2 x - 5)$
$x = - y$ , $- x = y$	$\frac{1}{2} (x - 3) = - (2 x - 5)$

2. Solve the two equations for x

26 additional steps

$\frac{1}{2} \cdot (x - 3) = (2 x - 5)$

Multiply the fraction(s):

$\frac{(1 \cdot (x - 3))}{2} = (2 x - 5)$

Break up the fraction:

$\frac{x}{2} + \frac{- 3}{2} = (2 x - 5)$

Subtract from both sides:

$(\frac{x}{2} + \frac{- 3}{2}) - 2 x = (2 x - 5) - 2 x$

Group like terms:

$(\frac{x}{2} - 2 x) + \frac{- 3}{2} = (2 x - 5) - 2 x$

Group the coefficients:

$(\frac{1}{2} - 2) x + \frac{- 3}{2} = (2 x - 5) - 2 x$

Convert the integer into a fraction:

$(\frac{1}{2} + \frac{- 4}{2}) x + \frac{- 3}{2} = (2 x - 5) - 2 x$

Combine the fractions:

$\frac{(1 - 4)}{2} x + \frac{- 3}{2} = (2 x - 5) - 2 x$

Combine the numerators:

$\frac{- 3}{2} x + \frac{- 3}{2} = (2 x - 5) - 2 x$

Group like terms:

$\frac{- 3}{2} x + \frac{- 3}{2} = (2 x - 2 x) - 5$

Simplify the arithmetic:

$\frac{- 3}{2} x + \frac{- 3}{2} = - 5$

Add to both sides:

$(\frac{- 3}{2} x + \frac{- 3}{2}) + \frac{3}{2} = - 5 + \frac{3}{2}$

Combine the fractions:

$\frac{- 3}{2} x + \frac{(- 3 + 3)}{2} = - 5 + \frac{3}{2}$

Combine the numerators:

$\frac{- 3}{2} x + \frac{0}{2} = - 5 + \frac{3}{2}$

Reduce the zero numerator:

$\frac{- 3}{2} x + 0 = - 5 + \frac{3}{2}$

Simplify the arithmetic:

$\frac{- 3}{2} x = - 5 + \frac{3}{2}$

Convert the integer into a fraction:

$\frac{- 3}{2} x = \frac{- 10}{2} + \frac{3}{2}$

Combine the fractions:

$\frac{- 3}{2} x = \frac{(- 10 + 3)}{2}$

Combine the numerators:

$\frac{- 3}{2} x = \frac{- 7}{2}$

Multiply both sides by inverse fraction :

$(\frac{- 3}{2} x) \cdot \frac{2}{- 3} = (\frac{- 7}{2}) \cdot \frac{2}{- 3}$

Move the negative sign from the denominator to the numerator:

$\frac{- 3}{2} x \cdot \frac{- 2}{3} = (\frac{- 7}{2}) \cdot \frac{2}{- 3}$

Group like terms:

$(\frac{- 3}{2} \cdot \frac{- 2}{3}) x = (\frac{- 7}{2}) \cdot \frac{2}{- 3}$

Multiply the coefficients:

$\frac{(- 3 \cdot - 2)}{(2 \cdot 3)} x = (\frac{- 7}{2}) \cdot \frac{2}{- 3}$

Simplify the arithmetic:

$1 x = (\frac{- 7}{2}) \cdot \frac{2}{- 3}$

$x = (\frac{- 7}{2}) \cdot \frac{2}{- 3}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 7}{2} \cdot \frac{- 2}{3}$

Multiply the fraction(s):

$x = \frac{(- 7 \cdot - 2)}{(2 \cdot 3)}$

Simplify the arithmetic:

$x = \frac{7}{3}$

24 additional steps

$\frac{1}{2} \cdot (x - 3) = - (2 x - 5)$

Multiply the fraction(s):

$\frac{(1 \cdot (x - 3))}{2} = - (2 x - 5)$

Break up the fraction:

$\frac{x}{2} + \frac{- 3}{2} = - (2 x - 5)$

Expand the parentheses:

$\frac{x}{2} + \frac{- 3}{2} = - 2 x + 5$

Add to both sides:

$(\frac{x}{2} + \frac{- 3}{2}) + 2 x = (- 2 x + 5) + 2 x$

Group like terms:

$(\frac{x}{2} + 2 x) + \frac{- 3}{2} = (- 2 x + 5) + 2 x$

Group the coefficients:

$(\frac{1}{2} + 2) x + \frac{- 3}{2} = (- 2 x + 5) + 2 x$

Convert the integer into a fraction:

$(\frac{1}{2} + \frac{4}{2}) x + \frac{- 3}{2} = (- 2 x + 5) + 2 x$

Combine the fractions:

$\frac{(1 + 4)}{2} x + \frac{- 3}{2} = (- 2 x + 5) + 2 x$

Combine the numerators:

$\frac{5}{2} x + \frac{- 3}{2} = (- 2 x + 5) + 2 x$

Group like terms:

$\frac{5}{2} x + \frac{- 3}{2} = (- 2 x + 2 x) + 5$

Simplify the arithmetic:

$\frac{5}{2} x + \frac{- 3}{2} = 5$

Add to both sides:

$(\frac{5}{2} x + \frac{- 3}{2}) + \frac{3}{2} = 5 + \frac{3}{2}$

Combine the fractions:

$\frac{5}{2} x + \frac{(- 3 + 3)}{2} = 5 + \frac{3}{2}$

Combine the numerators:

$\frac{5}{2} x + \frac{0}{2} = 5 + \frac{3}{2}$

Reduce the zero numerator:

$\frac{5}{2} x + 0 = 5 + \frac{3}{2}$

Simplify the arithmetic:

$\frac{5}{2} x = 5 + \frac{3}{2}$

Convert the integer into a fraction:

$\frac{5}{2} x = \frac{10}{2} + \frac{3}{2}$

Combine the fractions:

$\frac{5}{2} x = \frac{(10 + 3)}{2}$

Combine the numerators:

$\frac{5}{2} x = \frac{13}{2}$

Multiply both sides by inverse fraction :

$(\frac{5}{2} x) \cdot \frac{2}{5} = (\frac{13}{2}) \cdot \frac{2}{5}$

Group like terms:

$(\frac{5}{2} \cdot \frac{2}{5}) x = (\frac{13}{2}) \cdot \frac{2}{5}$

Multiply the coefficients:

$\frac{(5 \cdot 2)}{(2 \cdot 5)} x = (\frac{13}{2}) \cdot \frac{2}{5}$

Simplify the fraction:

$x = (\frac{13}{2}) \cdot \frac{2}{5}$

Multiply the fraction(s):

$x = \frac{(13 \cdot 2)}{(2 \cdot 5)}$

Simplify the arithmetic:

$x = \frac{13}{5}$

3. List the solutions

$x = \frac{7}{3}, \frac{13}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = \frac{1}{2} | x - 3 |$
$y = | 2 x - 5 |$
The equation is true where the two lines cross.

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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.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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