Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = 16, \frac{9}{7}

x=16 , \frac{9}{7}

Mixed number form:

x = 16, 1 \frac{2}{7}

x=16 , 1\frac{2}{7}

Decimal form:

x = 16, 1.286

x=16 , 1.286

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
$| 6 x + 7 | = | 8 x - 25 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 x + 7 \| = \| 8 x - 25 \|$
$x = + y$	$(6 x + 7) = (8 x - 25)$
$x = - y$	$(6 x + 7) = - (8 x - 25)$
$+ x = y$	$(6 x + 7) = (8 x - 25)$
$- x = y$	$- (6 x + 7) = (8 x - 25)$

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:

$\| x \| = \| y \|$	$\| 6 x + 7 \| = \| 8 x - 25 \|$
$x = + y$ , $+ x = y$	$(6 x + 7) = (8 x - 25)$
$x = - y$ , $- x = y$	$(6 x + 7) = - (8 x - 25)$

2. Solve the two equations for x

13 additional steps

$(6 x + 7) = (8 x - 25)$

Subtract from both sides:

$(6 x + 7) - 8 x = (8 x - 25) - 8 x$

Group like terms:

$(6 x - 8 x) + 7 = (8 x - 25) - 8 x$

Simplify the arithmetic:

$- 2 x + 7 = (8 x - 25) - 8 x$

Group like terms:

$- 2 x + 7 = (8 x - 8 x) - 25$

Simplify the arithmetic:

$- 2 x + 7 = - 25$

Subtract from both sides:

$(- 2 x + 7) - 7 = - 25 - 7$

Simplify the arithmetic:

$- 2 x = - 25 - 7$

Simplify the arithmetic:

$- 2 x = - 32$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{- 32}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{- 32}{- 2}$

Simplify the fraction:

$x = \frac{- 32}{- 2}$

Cancel out the negatives:

$x = \frac{32}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(16 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 16$

12 additional steps

$(6 x + 7) = - (8 x - 25)$

Expand the parentheses:

$(6 x + 7) = - 8 x + 25$

Add to both sides:

$(6 x + 7) + 8 x = (- 8 x + 25) + 8 x$

Group like terms:

$(6 x + 8 x) + 7 = (- 8 x + 25) + 8 x$

Simplify the arithmetic:

$14 x + 7 = (- 8 x + 25) + 8 x$

Group like terms:

$14 x + 7 = (- 8 x + 8 x) + 25$

Simplify the arithmetic:

$14 x + 7 = 25$

Subtract from both sides:

$(14 x + 7) - 7 = 25 - 7$

Simplify the arithmetic:

$14 x = 25 - 7$

Simplify the arithmetic:

$14 x = 18$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{18}{14}$

Simplify the fraction:

$x = \frac{18}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(9 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{9}{7}$

3. List the solutions

$x = 16, \frac{9}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 x + 7 |$
$y = | 8 x - 25 |$
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