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Solution - Absolute value equations

Exact form:

x = - \frac{31}{10}, \frac{1}{130}

x=-\frac{31}{10} , \frac{1}{130}

Mixed number form:

x = - 3 \frac{1}{10}, \frac{1}{130}

x=-3\frac{1}{10} , \frac{1}{130}

Decimal form:

x = - 3.1, 0.008

x=-3.1 , 0.008

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
$| 6 x - \frac{8}{5} | = | 7 x + \frac{3}{2} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 x - \frac{8}{5} \| = \| 7 x + \frac{3}{2} \|$
$x = + y$	$(6 x - \frac{8}{5}) = (7 x + \frac{3}{2})$
$x = - y$	$(6 x - \frac{8}{5}) = - (7 x + \frac{3}{2})$
$+ x = y$	$(6 x - \frac{8}{5}) = (7 x + \frac{3}{2})$
$- x = y$	$- (6 x - \frac{8}{5}) = (7 x + \frac{3}{2})$

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:

$\| x \| = \| y \|$	$\| 6 x - \frac{8}{5} \| = \| 7 x + \frac{3}{2} \|$
$x = + y$ , $+ x = y$	$(6 x - \frac{8}{5}) = (7 x + \frac{3}{2})$
$x = - y$ , $- x = y$	$(6 x - \frac{8}{5}) = - (7 x + \frac{3}{2})$

2. Solve the two equations for x

17 additional steps

$(6 x + \frac{- 8}{5}) = (7 x + \frac{3}{2})$

Subtract from both sides:

$(6 x + \frac{- 8}{5}) - 7 x = (7 x + \frac{3}{2}) - 7 x$

Group like terms:

$(6 x - 7 x) + \frac{- 8}{5} = (7 x + \frac{3}{2}) - 7 x$

Simplify the arithmetic:

$- x + \frac{- 8}{5} = (7 x + \frac{3}{2}) - 7 x$

Group like terms:

$- x + \frac{- 8}{5} = (7 x - 7 x) + \frac{3}{2}$

Simplify the arithmetic:

$- x + \frac{- 8}{5} = \frac{3}{2}$

Add to both sides:

$(- x + \frac{- 8}{5}) + \frac{8}{5} = (\frac{3}{2}) + \frac{8}{5}$

Combine the fractions:

$- x + \frac{(- 8 + 8)}{5} = (\frac{3}{2}) + \frac{8}{5}$

Combine the numerators:

$- x + \frac{0}{5} = (\frac{3}{2}) + \frac{8}{5}$

Reduce the zero numerator:

$- x + 0 = (\frac{3}{2}) + \frac{8}{5}$

Simplify the arithmetic:

$- x = (\frac{3}{2}) + \frac{8}{5}$

Find the lowest common denominator:

$- x = \frac{(3 \cdot 5)}{(2 \cdot 5)} + \frac{(8 \cdot 2)}{(5 \cdot 2)}$

Multiply the denominators:

$- x = \frac{(3 \cdot 5)}{10} + \frac{(8 \cdot 2)}{10}$

Multiply the numerators:

$- x = \frac{15}{10} + \frac{16}{10}$

Combine the fractions:

$- x = \frac{(15 + 16)}{10}$

Combine the numerators:

$- x = \frac{31}{10}$

Multiply both sides by :

$- x \cdot - 1 = (\frac{31}{10}) \cdot - 1$

Remove the one(s):

$x = (\frac{31}{10}) \cdot - 1$

Remove the one(s):

$x = \frac{- 31}{10}$

19 additional steps

$(6 x + \frac{- 8}{5}) = - (7 x + \frac{3}{2})$

Expand the parentheses:

$(6 x + \frac{- 8}{5}) = - 7 x + \frac{- 3}{2}$

Add to both sides:

$(6 x + \frac{- 8}{5}) + 7 x = (- 7 x + \frac{- 3}{2}) + 7 x$

Group like terms:

$(6 x + 7 x) + \frac{- 8}{5} = (- 7 x + \frac{- 3}{2}) + 7 x$

Simplify the arithmetic:

$13 x + \frac{- 8}{5} = (- 7 x + \frac{- 3}{2}) + 7 x$

Group like terms:

$13 x + \frac{- 8}{5} = (- 7 x + 7 x) + \frac{- 3}{2}$

Simplify the arithmetic:

$13 x + \frac{- 8}{5} = \frac{- 3}{2}$

Add to both sides:

$(13 x + \frac{- 8}{5}) + \frac{8}{5} = (\frac{- 3}{2}) + \frac{8}{5}$

Combine the fractions:

$13 x + \frac{(- 8 + 8)}{5} = (\frac{- 3}{2}) + \frac{8}{5}$

Combine the numerators:

$13 x + \frac{0}{5} = (\frac{- 3}{2}) + \frac{8}{5}$

Reduce the zero numerator:

$13 x + 0 = (\frac{- 3}{2}) + \frac{8}{5}$

Simplify the arithmetic:

$13 x = (\frac{- 3}{2}) + \frac{8}{5}$

Find the lowest common denominator:

$13 x = \frac{(- 3 \cdot 5)}{(2 \cdot 5)} + \frac{(8 \cdot 2)}{(5 \cdot 2)}$

Multiply the denominators:

$13 x = \frac{(- 3 \cdot 5)}{10} + \frac{(8 \cdot 2)}{10}$

Multiply the numerators:

$13 x = \frac{- 15}{10} + \frac{16}{10}$

Combine the fractions:

$13 x = \frac{(- 15 + 16)}{10}$

Combine the numerators:

$13 x = \frac{1}{10}$

Divide both sides by :

$\frac{(13 x)}{13} = \frac{(\frac{1}{10})}{13}$

Simplify the fraction:

$x = \frac{(\frac{1}{10})}{13}$

Simplify the arithmetic:

$x = \frac{1}{(10 \cdot 13)}$

$x = \frac{1}{130}$

3. List the solutions

$x = - \frac{31}{10}, \frac{1}{130}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 x - \frac{8}{5} |$
$y = | 7 x + \frac{3}{2} |$
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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