Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

v = - \frac{3}{2}

v=-\frac{3}{2}

Mixed number form:

v = - 1 \frac{1}{2}

v=-1\frac{1}{2}

Decimal form:

v = - 1.5

v=-1.5

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
$| 5 v + 7 | = | 5 v + 8 |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for v

5 additional steps

$(5 v + 7) = (5 v + 8)$

Subtract from both sides:

$(5 v + 7) - 5 v = (5 v + 8) - 5 v$

Group like terms:

$(5 v - 5 v) + 7 = (5 v + 8) - 5 v$

Simplify the arithmetic:

$7 = (5 v + 8) - 5 v$

Group like terms:

$7 = (5 v - 5 v) + 8$

Simplify the arithmetic:

$7 = 8$

The statement is false:

$7 = 8$

The equation is false so it has no solution.

12 additional steps

$(5 v + 7) = - (5 v + 8)$

Expand the parentheses:

$(5 v + 7) = - 5 v - 8$

Add to both sides:

$(5 v + 7) + 5 v = (- 5 v - 8) + 5 v$

Group like terms:

$(5 v + 5 v) + 7 = (- 5 v - 8) + 5 v$

Simplify the arithmetic:

$10 v + 7 = (- 5 v - 8) + 5 v$

Group like terms:

$10 v + 7 = (- 5 v + 5 v) - 8$

Simplify the arithmetic:

$10 v + 7 = - 8$

Subtract from both sides:

$(10 v + 7) - 7 = - 8 - 7$

Simplify the arithmetic:

$10 v = - 8 - 7$

Simplify the arithmetic:

$10 v = - 15$

Divide both sides by :

$\frac{(10 v)}{10} = \frac{- 15}{10}$

Simplify the fraction:

$v = \frac{- 15}{10}$

Find the greatest common factor of the numerator and denominator:

$v = \frac{(- 3 \cdot 5)}{(2 \cdot 5)}$

Factor out and cancel the greatest common factor:

$v = \frac{- 3}{2}$

3. Graph

Each line represents the function of one side of the equation:
$y = | 5 v + 7 |$
$y = | 5 v + 8 |$
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 v

3. 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 v

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved