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Solution - Absolute value equations

Exact form:

x = - \frac{3}{8}, - \frac{5}{4}

x=-\frac{3}{8} , -\frac{5}{4}

Mixed number form:

x = - \frac{3}{8}, - 1 \frac{1}{4}

x=-\frac{3}{8} , -1\frac{1}{4}

Decimal form:

x = - 0.375, - 1.25

x=-0.375 , -1.25

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

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

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:

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

2. Solve the two equations for x

17 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Combine the fractions:

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

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Simplify the arithmetic:

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

$x = \frac{- 3}{8}$

17 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Combine the fractions:

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

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Divide both sides by :

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

Simplify the fraction:

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

Simplify the arithmetic:

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

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

3. List the solutions

$x = - \frac{3}{8}, - \frac{5}{4}$
(2 solution(s))

4. Graph

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