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Solution - Absolute value equations

Exact form:

x = - 24, \frac{12}{7}

x=-24 , \frac{12}{7}

Mixed number form:

x = - 24, 1 \frac{5}{7}

x=-24 , 1\frac{5}{7}

Decimal form:

x = - 24, 1.714

x=-24 , 1.714

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

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

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:

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

2. Solve the two equations for x

21 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

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

Multiply the numerators:

$(\frac{3}{6} + \frac{- 4}{6}) x - 3 = (\frac{2}{3} \cdot x + 1) - \frac{2}{3} x$

Combine the fractions:

$\frac{(3 - 4)}{6} \cdot x - 3 = (\frac{2}{3} \cdot x + 1) - \frac{2}{3} x$

Combine the numerators:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

$\frac{- 1}{6} \cdot x - 3 = \frac{0}{3} x + 1$

Reduce the zero numerator:

$\frac{- 1}{6} x - 3 = 0 x + 1$

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$\frac{- 1}{6} x = 4$

Multiply both sides by inverse fraction :

$(\frac{- 1}{6} x) \cdot \frac{6}{- 1} = 4 \cdot \frac{6}{- 1}$

Group like terms:

$(\frac{- 1}{6} \cdot - 6) x = 4 \cdot \frac{6}{- 1}$

Multiply the coefficients:

$\frac{(- 1 \cdot - 6)}{6} x = 4 \cdot \frac{6}{- 1}$

Simplify the arithmetic:

$1 x = 4 \cdot \frac{6}{- 1}$

$x = 4 \cdot \frac{6}{- 1}$

Simplify the arithmetic:

$x = - 24$

22 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

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

Multiply the numerators:

$(\frac{3}{6} + \frac{4}{6}) x - 3 = (\frac{- 2}{3} \cdot x - 1) + \frac{2}{3} x$

Combine the fractions:

$\frac{(3 + 4)}{6} \cdot x - 3 = (\frac{- 2}{3} \cdot x - 1) + \frac{2}{3} x$

Combine the numerators:

$\frac{7}{6} \cdot x - 3 = (\frac{- 2}{3} \cdot x - 1) + \frac{2}{3} x$

Group like terms:

$\frac{7}{6} \cdot x - 3 = (\frac{- 2}{3} \cdot x + \frac{2}{3} x) - 1$

Combine the fractions:

$\frac{7}{6} \cdot x - 3 = \frac{(- 2 + 2)}{3} x - 1$

Combine the numerators:

$\frac{7}{6} \cdot x - 3 = \frac{0}{3} x - 1$

Reduce the zero numerator:

$\frac{7}{6} x - 3 = 0 x - 1$

Simplify the arithmetic:

$\frac{7}{6} x - 3 = - 1$

Add to both sides:

$(\frac{7}{6} x - 3) + 3 = - 1 + 3$

Simplify the arithmetic:

$\frac{7}{6} x = - 1 + 3$

Simplify the arithmetic:

$\frac{7}{6} x = 2$

Multiply both sides by inverse fraction :

$(\frac{7}{6} x) \cdot \frac{6}{7} = 2 \cdot \frac{6}{7}$

Group like terms:

$(\frac{7}{6} \cdot \frac{6}{7}) x = 2 \cdot \frac{6}{7}$

Multiply the coefficients:

$\frac{(7 \cdot 6)}{(6 \cdot 7)} x = 2 \cdot \frac{6}{7}$

Simplify the fraction:

$x = 2 \cdot \frac{6}{7}$

Multiply the fraction(s):

$x = \frac{(2 \cdot 6)}{7}$

Simplify the arithmetic:

$x = \frac{12}{7}$

3. List the solutions

$x = - 24, \frac{12}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{1}{2} x - 3 |$
$y = | \frac{2}{3} x + 1 |$
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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