Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - \frac{8}{5}, \frac{8}{65}

x=-\frac{8}{5} , \frac{8}{65}

Mixed number form:

x = - 1 \frac{3}{5}, \frac{8}{65}

x=-1\frac{3}{5} , \frac{8}{65}

Decimal form:

x = - 1.6, 0.123

x=-1.6 , 0.123

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

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

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:

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

2. Solve the two equations for x

12 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$- x + \frac{- 8}{5} = 0$

Add to both sides:

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

Combine the fractions:

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

Combine the numerators:

$- x + \frac{0}{5} = 0 + \frac{8}{5}$

Reduce the zero numerator:

$- x + 0 = 0 + \frac{8}{5}$

Simplify the arithmetic:

$- x = 0 + \frac{8}{5}$

Simplify the arithmetic:

$- x = \frac{8}{5}$

Multiply both sides by :

$- x \cdot - 1 = (\frac{8}{5}) \cdot - 1$

Remove the one(s):

$x = (\frac{8}{5}) \cdot - 1$

Remove the one(s):

$x = \frac{- 8}{5}$

12 additional steps

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

Add to both sides:

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

Combine the fractions:

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

Combine the numerators:

$6 x + \frac{0}{5} = (- 7 x) + \frac{8}{5}$

Reduce the zero numerator:

$6 x + 0 = (- 7 x) + \frac{8}{5}$

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$13 x = \frac{8}{5}$

Divide both sides by :

$\frac{(13 x)}{13} = \frac{(\frac{8}{5})}{13}$

Simplify the fraction:

$x = \frac{(\frac{8}{5})}{13}$

Simplify the arithmetic:

$x = \frac{8}{(5 \cdot 13)}$

$x = \frac{8}{65}$

3. List the solutions

$x = - \frac{8}{5}, \frac{8}{65}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 x - \frac{8}{5} |$
$y = | 7 x |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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

Latest Related Drills Solved