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Solution - Absolute value equations

Exact form:

x = 16, 0

x=16 , 0

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
$| \frac{1}{2} x + 2 | = | \frac{3}{4} x - 2 |$
without the absolute value bars:

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

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:

$\| x \| = \| y \|$	$\| \frac{1}{2} x + 2 \| = \| \frac{3}{4} x - 2 \|$
$x = + y$ , $+ x = y$	$(\frac{1}{2} x + 2) = (\frac{3}{4} x - 2)$
$x = - y$ , $- x = y$	$(\frac{1}{2} x + 2) = - (\frac{3}{4} x - 2)$

2. Solve the two equations for x

21 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

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

Multiply the numerators:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

$\frac{- 1}{4} \cdot x + 2 = \frac{0}{4} x - 2$

Reduce the zero numerator:

$\frac{- 1}{4} x + 2 = 0 x - 2$

Simplify the arithmetic:

$\frac{- 1}{4} x + 2 = - 2$

Subtract from both sides:

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

Simplify the arithmetic:

$\frac{- 1}{4} x = - 2 - 2$

Simplify the arithmetic:

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

Multiply both sides by inverse fraction :

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

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

Simplify the arithmetic:

$x = 16$

17 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

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

Multiply the numerators:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

$\frac{5}{4} \cdot x + 2 = \frac{0}{4} x + 2$

Reduce the zero numerator:

$\frac{5}{4} x + 2 = 0 x + 2$

Simplify the arithmetic:

$\frac{5}{4} x + 2 = 2$

Subtract from both sides:

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

Simplify the arithmetic:

$\frac{5}{4} x = 2 - 2$

Simplify the arithmetic:

$\frac{5}{4} x = 0$

Divide both sides by the coefficient:

$x = 0$

3. List the solutions

$x = 16, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{1}{2} x + 2 |$
$y = | \frac{3}{4} x - 2 |$
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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