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Solution - Absolute value equations

Exact form:

x = - \frac{15}{2}, \frac{21}{10}

x=-\frac{15}{2} , \frac{21}{10}

Mixed number form:

x = - 7 \frac{1}{2}, 2 \frac{1}{10}

x=-7\frac{1}{2} , 2\frac{1}{10}

Decimal form:

x = - 7.5, 2.1

x=-7.5 , 2.1

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
$| \frac{2}{3} x - 3 | = | x - \frac{1}{2} |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for x

19 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Simplify the arithmetic:

$\frac{- 1}{3} x - 3 = \frac{- 1}{2}$

Add to both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

$\frac{- 1}{3} x = \frac{- 1}{2} + \frac{6}{2}$

Combine the fractions:

$\frac{- 1}{3} x = \frac{(- 1 + 6)}{2}$

Combine the numerators:

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

Multiply both sides by inverse fraction :

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

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

Multiply the fraction(s):

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

Simplify the arithmetic:

$x = \frac{- 15}{2}$

20 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

$\frac{5}{3} x = \frac{1}{2} + \frac{6}{2}$

Combine the fractions:

$\frac{5}{3} x = \frac{(1 + 6)}{2}$

Combine the numerators:

$\frac{5}{3} x = \frac{7}{2}$

Multiply both sides by inverse fraction :

$(\frac{5}{3} x) \cdot \frac{3}{5} = (\frac{7}{2}) \cdot \frac{3}{5}$

Group like terms:

$(\frac{5}{3} \cdot \frac{3}{5}) x = (\frac{7}{2}) \cdot \frac{3}{5}$

Multiply the coefficients:

$\frac{(5 \cdot 3)}{(3 \cdot 5)} x = (\frac{7}{2}) \cdot \frac{3}{5}$

Simplify the fraction:

$x = (\frac{7}{2}) \cdot \frac{3}{5}$

Multiply the fraction(s):

$x = \frac{(7 \cdot 3)}{(2 \cdot 5)}$

Simplify the arithmetic:

$x = \frac{21}{(2 \cdot 5)}$

$x = \frac{21}{10}$

3. List the solutions

$x = - \frac{15}{2}, \frac{21}{10}$
(2 solution(s))

4. Graph

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

How did we do?

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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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