Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

u = \frac{1}{17}, \frac{17}{3}

u=\frac{1}{17} , \frac{17}{3}

Mixed number form:

u = \frac{1}{17}, 5 \frac{2}{3}

u=\frac{1}{17} , 5\frac{2}{3}

Decimal form:

u = 0.059, 5.667

u=0.059 , 5.667

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
$| 7 u + 8 | = | - 10 u + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 u + 8 \| = \| - 10 u + 9 \|$
$x = + y$	$(7 u + 8) = (- 10 u + 9)$
$x = - y$	$(7 u + 8) = - (- 10 u + 9)$
$+ x = y$	$(7 u + 8) = (- 10 u + 9)$
$- x = y$	$- (7 u + 8) = (- 10 u + 9)$

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:

$\| x \| = \| y \|$	$\| 7 u + 8 \| = \| - 10 u + 9 \|$
$x = + y$ , $+ x = y$	$(7 u + 8) = (- 10 u + 9)$
$x = - y$ , $- x = y$	$(7 u + 8) = - (- 10 u + 9)$

2. Solve the two equations for u

9 additional steps

$(7 u + 8) = (- 10 u + 9)$

Add to both sides:

$(7 u + 8) + 10 u = (- 10 u + 9) + 10 u$

Group like terms:

$(7 u + 10 u) + 8 = (- 10 u + 9) + 10 u$

Simplify the arithmetic:

$17 u + 8 = (- 10 u + 9) + 10 u$

Group like terms:

$17 u + 8 = (- 10 u + 10 u) + 9$

Simplify the arithmetic:

$17 u + 8 = 9$

Subtract from both sides:

$(17 u + 8) - 8 = 9 - 8$

Simplify the arithmetic:

$17 u = 9 - 8$

Simplify the arithmetic:

$17 u = 1$

Divide both sides by :

$\frac{(17 u)}{17} = \frac{1}{17}$

Simplify the fraction:

$u = \frac{1}{17}$

12 additional steps

$(7 u + 8) = - (- 10 u + 9)$

Expand the parentheses:

$(7 u + 8) = 10 u - 9$

Subtract from both sides:

$(7 u + 8) - 10 u = (10 u - 9) - 10 u$

Group like terms:

$(7 u - 10 u) + 8 = (10 u - 9) - 10 u$

Simplify the arithmetic:

$- 3 u + 8 = (10 u - 9) - 10 u$

Group like terms:

$- 3 u + 8 = (10 u - 10 u) - 9$

Simplify the arithmetic:

$- 3 u + 8 = - 9$

Subtract from both sides:

$(- 3 u + 8) - 8 = - 9 - 8$

Simplify the arithmetic:

$- 3 u = - 9 - 8$

Simplify the arithmetic:

$- 3 u = - 17$

Divide both sides by :

$\frac{(- 3 u)}{- 3} = \frac{- 17}{- 3}$

Cancel out the negatives:

$\frac{3 u}{3} = \frac{- 17}{- 3}$

Simplify the fraction:

$u = \frac{- 17}{- 3}$

Cancel out the negatives:

$u = \frac{17}{3}$

3. List the solutions

$u = \frac{1}{17}, \frac{17}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 7 u + 8 |$
$y = | - 10 u + 9 |$
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 u

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 u

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved