Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

q = \frac{2}{7}, \frac{4}{3}

q=\frac{2}{7} , \frac{4}{3}

Mixed number form:

q = \frac{2}{7}, 1 \frac{1}{3}

q=\frac{2}{7} , 1\frac{1}{3}

Decimal form:

q = 0.286, 1.333

q=0.286 , 1.333

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
$- | 5 q - 3 | = | 2 q + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$- \| 5 q - 3 \| = \| 2 q + 1 \|$
$x = + y$	$- (5 q - 3) = (2 q + 1)$
$x = - y$	$- (5 q - 3) = - (2 q + 1)$
$+ x = y$	$- (5 q - 3) = (2 q + 1)$
$- x = y$	$- (- (5 q - 3)) = (2 q + 1)$

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:

$\| x \| = \| y \|$	$- \| 5 q - 3 \| = \| 2 q + 1 \|$
$x = + y$ , $+ x = y$	$- (5 q - 3) = (2 q + 1)$
$x = - y$ , $- x = y$	$- (5 q - 3) = - (2 q + 1)$

2. Solve the two equations for q

12 additional steps

$- (5 q - 3) = (2 q + 1)$

Expand the parentheses:

$- 5 q + 3 = (2 q + 1)$

Subtract from both sides:

$(- 5 q + 3) - 2 q = (2 q + 1) - 2 q$

Group like terms:

$(- 5 q - 2 q) + 3 = (2 q + 1) - 2 q$

Simplify the arithmetic:

$- 7 q + 3 = (2 q + 1) - 2 q$

Group like terms:

$- 7 q + 3 = (2 q - 2 q) + 1$

Simplify the arithmetic:

$- 7 q + 3 = 1$

Subtract from both sides:

$(- 7 q + 3) - 3 = 1 - 3$

Simplify the arithmetic:

$- 7 q = 1 - 3$

Simplify the arithmetic:

$- 7 q = - 2$

Divide both sides by :

$\frac{(- 7 q)}{- 7} = \frac{- 2}{- 7}$

Cancel out the negatives:

$\frac{7 q}{7} = \frac{- 2}{- 7}$

Simplify the fraction:

$q = \frac{- 2}{- 7}$

Cancel out the negatives:

$q = \frac{2}{7}$

13 additional steps

$- (5 q - 3) = - (2 q + 1)$

Expand the parentheses:

$- 5 q + 3 = - (2 q + 1)$

Expand the parentheses:

$- 5 q + 3 = - 2 q - 1$

Add to both sides:

$(- 5 q + 3) + 2 q = (- 2 q - 1) + 2 q$

Group like terms:

$(- 5 q + 2 q) + 3 = (- 2 q - 1) + 2 q$

Simplify the arithmetic:

$- 3 q + 3 = (- 2 q - 1) + 2 q$

Group like terms:

$- 3 q + 3 = (- 2 q + 2 q) - 1$

Simplify the arithmetic:

$- 3 q + 3 = - 1$

Subtract from both sides:

$(- 3 q + 3) - 3 = - 1 - 3$

Simplify the arithmetic:

$- 3 q = - 1 - 3$

Simplify the arithmetic:

$- 3 q = - 4$

Divide both sides by :

$\frac{(- 3 q)}{- 3} = \frac{- 4}{- 3}$

Cancel out the negatives:

$\frac{3 q}{3} = \frac{- 4}{- 3}$

Simplify the fraction:

$q = \frac{- 4}{- 3}$

Cancel out the negatives:

$q = \frac{4}{3}$

3. List the solutions

$q = \frac{2}{7}, \frac{4}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = - | 5 q - 3 |$
$y = | 2 q + 1 |$
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 q

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 q

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved