Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{59}{72}, \frac{11}{8}

x=\frac{59}{72} , \frac{11}{8}

Mixed number form:

x = \frac{59}{72}, 1 \frac{3}{8}

x=\frac{59}{72} , 1\frac{3}{8}

Decimal form:

x = 0.819, 1.375

x=0.819 , 1.375

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
$4 | 8 x - 6 | = 5 | - 8 x + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| 8 x - 6 \| = 5 \| - 8 x + 7 \|$
$x = + y$	$4 (8 x - 6) = 5 (- 8 x + 7)$
$x = - y$	$4 (8 x - 6) = 5 (- (- 8 x + 7))$
$+ x = y$	$4 (8 x - 6) = 5 (- 8 x + 7)$
$- x = y$	$4 (- (8 x - 6)) = 5 (- 8 x + 7)$

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:

$\| x \| = \| y \|$	$4 \| 8 x - 6 \| = 5 \| - 8 x + 7 \|$
$x = + y$ , $+ x = y$	$4 (8 x - 6) = 5 (- 8 x + 7)$
$x = - y$ , $- x = y$	$4 (8 x - 6) = 5 (- (- 8 x + 7))$

2. Solve the two equations for x

15 additional steps

$4 \cdot (8 x - 6) = 5 \cdot (- 8 x + 7)$

Expand the parentheses:

$4 \cdot 8 x + 4 \cdot - 6 = 5 \cdot (- 8 x + 7)$

Multiply the coefficients:

$32 x + 4 \cdot - 6 = 5 \cdot (- 8 x + 7)$

Simplify the arithmetic:

$32 x - 24 = 5 \cdot (- 8 x + 7)$

Expand the parentheses:

$32 x - 24 = 5 \cdot - 8 x + 5 \cdot 7$

Multiply the coefficients:

$32 x - 24 = - 40 x + 5 \cdot 7$

Simplify the arithmetic:

$32 x - 24 = - 40 x + 35$

Add to both sides:

$(32 x - 24) + 40 x = (- 40 x + 35) + 40 x$

Group like terms:

$(32 x + 40 x) - 24 = (- 40 x + 35) + 40 x$

Simplify the arithmetic:

$72 x - 24 = (- 40 x + 35) + 40 x$

Group like terms:

$72 x - 24 = (- 40 x + 40 x) + 35$

Simplify the arithmetic:

$72 x - 24 = 35$

Add to both sides:

$(72 x - 24) + 24 = 35 + 24$

Simplify the arithmetic:

$72 x = 35 + 24$

Simplify the arithmetic:

$72 x = 59$

Divide both sides by :

$\frac{(72 x)}{72} = \frac{59}{72}$

Simplify the fraction:

$x = \frac{59}{72}$

18 additional steps

$4 \cdot (8 x - 6) = 5 \cdot (- (- 8 x + 7))$

Expand the parentheses:

$4 \cdot 8 x + 4 \cdot - 6 = 5 \cdot (- (- 8 x + 7))$

Multiply the coefficients:

$32 x + 4 \cdot - 6 = 5 \cdot (- (- 8 x + 7))$

Simplify the arithmetic:

$32 x - 24 = 5 \cdot (- (- 8 x + 7))$

Expand the parentheses:

$32 x - 24 = 5 \cdot (8 x - 7)$

Expand the parentheses:

$32 x - 24 = 5 \cdot 8 x + 5 \cdot - 7$

Multiply the coefficients:

$32 x - 24 = 40 x + 5 \cdot - 7$

Simplify the arithmetic:

$32 x - 24 = 40 x - 35$

Subtract from both sides:

$(32 x - 24) - 40 x = (40 x - 35) - 40 x$

Group like terms:

$(32 x - 40 x) - 24 = (40 x - 35) - 40 x$

Simplify the arithmetic:

$- 8 x - 24 = (40 x - 35) - 40 x$

Group like terms:

$- 8 x - 24 = (40 x - 40 x) - 35$

Simplify the arithmetic:

$- 8 x - 24 = - 35$

Add to both sides:

$(- 8 x - 24) + 24 = - 35 + 24$

Simplify the arithmetic:

$- 8 x = - 35 + 24$

Simplify the arithmetic:

$- 8 x = - 11$

Divide both sides by :

$\frac{(- 8 x)}{- 8} = \frac{- 11}{- 8}$

Cancel out the negatives:

$\frac{8 x}{8} = \frac{- 11}{- 8}$

Simplify the fraction:

$x = \frac{- 11}{- 8}$

Cancel out the negatives:

$x = \frac{11}{8}$

3. List the solutions

$x = \frac{59}{72}, \frac{11}{8}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 4 | 8 x - 6 |$
$y = 5 | - 8 x + 7 |$
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