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Solution - Absolute value equations

Exact form:

x = \frac{5}{16}, \frac{7}{8}

x=\frac{5}{16} , \frac{7}{8}

Decimal form:

x = 0.312, 0.875

x=0.312 , 0.875

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

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

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:

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

2. Solve the two equations for x

18 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Combine the fractions:

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

Combine the numerators:

$4 x + \frac{0}{4} = (\frac{3}{2}) - \frac{1}{4}$

Reduce the zero numerator:

$4 x + 0 = (\frac{3}{2}) - \frac{1}{4}$

Simplify the arithmetic:

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

Find the lowest common denominator:

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

Multiply the denominators:

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

Multiply the numerators:

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

Combine the fractions:

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

Combine the numerators:

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

Divide both sides by :

$\frac{(4 x)}{4} = \frac{(\frac{5}{4})}{4}$

Simplify the fraction:

$x = \frac{(\frac{5}{4})}{4}$

Simplify the arithmetic:

$x = \frac{5}{(4 \cdot 4)}$

$x = \frac{5}{16}$

20 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Combine the fractions:

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

Combine the numerators:

$- 2 x + \frac{0}{4} = (\frac{- 3}{2}) - \frac{1}{4}$

Reduce the zero numerator:

$- 2 x + 0 = (\frac{- 3}{2}) - \frac{1}{4}$

Simplify the arithmetic:

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

Find the lowest common denominator:

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

Multiply the denominators:

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

Multiply the numerators:

$- 2 x = \frac{- 6}{4} + \frac{- 1}{4}$

Combine the fractions:

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

Combine the numerators:

$- 2 x = \frac{- 7}{4}$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{(\frac{- 7}{4})}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{(\frac{- 7}{4})}{- 2}$

Simplify the fraction:

$x = \frac{(\frac{- 7}{4})}{- 2}$

Simplify the arithmetic:

$x = \frac{- 7}{(4 \cdot - 2)}$

$x = \frac{7}{8}$

3. List the solutions

$x = \frac{5}{16}, \frac{7}{8}$
(2 solution(s))

4. Graph

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