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Solution - Absolute value equations

Exact form:

x = - \frac{25}{8}, \frac{23}{16}

x=-\frac{25}{8} , \frac{23}{16}

Mixed number form:

x = - 3 \frac{1}{8}, 1 \frac{7}{16}

x=-3\frac{1}{8} , 1\frac{7}{16}

Decimal form:

x = - 3.125, 1.438

x=-3.125 , 1.438

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
$| 3 x + \frac{1}{4} | = | x - 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x + \frac{1}{4} \| = \| x - 6 \|$
$x = + y$	$(3 x + \frac{1}{4}) = (x - 6)$
$x = - y$	$(3 x + \frac{1}{4}) = - (x - 6)$
$+ x = y$	$(3 x + \frac{1}{4}) = (x - 6)$
$- x = y$	$- (3 x + \frac{1}{4}) = (x - 6)$

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:

$\| x \| = \| y \|$	$\| 3 x + \frac{1}{4} \| = \| x - 6 \|$
$x = + y$ , $+ x = y$	$(3 x + \frac{1}{4}) = (x - 6)$
$x = - y$ , $- x = y$	$(3 x + \frac{1}{4}) = - (x - 6)$

2. Solve the two equations for x

16 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Combine the fractions:

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

Combine the numerators:

$2 x + \frac{0}{4} = - 6 - \frac{1}{4}$

Reduce the zero numerator:

$2 x + 0 = - 6 - \frac{1}{4}$

Simplify the arithmetic:

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

Convert the integer into a fraction:

$2 x = \frac{- 24}{4} + \frac{- 1}{4}$

Combine the fractions:

$2 x = \frac{(- 24 - 1)}{4}$

Combine the numerators:

$2 x = \frac{- 25}{4}$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{(\frac{- 25}{4})}{2}$

Simplify the fraction:

$x = \frac{(\frac{- 25}{4})}{2}$

Simplify the arithmetic:

$x = \frac{- 25}{(4 \cdot 2)}$

$x = \frac{- 25}{8}$

17 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Combine the fractions:

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

Combine the numerators:

$4 x + \frac{0}{4} = 6 - \frac{1}{4}$

Reduce the zero numerator:

$4 x + 0 = 6 - \frac{1}{4}$

Simplify the arithmetic:

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

Convert the integer into a fraction:

$4 x = \frac{24}{4} + \frac{- 1}{4}$

Combine the fractions:

$4 x = \frac{(24 - 1)}{4}$

Combine the numerators:

$4 x = \frac{23}{4}$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{(\frac{23}{4})}{4}$

Simplify the fraction:

$x = \frac{(\frac{23}{4})}{4}$

Simplify the arithmetic:

$x = \frac{23}{(4 \cdot 4)}$

$x = \frac{23}{16}$

3. List the solutions

$x = - \frac{25}{8}, \frac{23}{16}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 x + \frac{1}{4} |$
$y = | x - 6 |$
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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