Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{17}{4}, - \frac{23}{6}

x=\frac{17}{4} , -\frac{23}{6}

Mixed number form:

x = 4 \frac{1}{4}, - 3 \frac{5}{6}

x=4\frac{1}{4} , -3\frac{5}{6}

Decimal form:

x = 4.25, - 3.833

x=4.25 , -3.833

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
$| 5 x + 3 | = | x + 20 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x + 3 \| = \| x + 20 \|$
$x = + y$	$(5 x + 3) = (x + 20)$
$x = - y$	$(5 x + 3) = - (x + 20)$
$+ x = y$	$(5 x + 3) = (x + 20)$
$- x = y$	$- (5 x + 3) = (x + 20)$

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:

$\| x \| = \| y \|$	$\| 5 x + 3 \| = \| x + 20 \|$
$x = + y$ , $+ x = y$	$(5 x + 3) = (x + 20)$
$x = - y$ , $- x = y$	$(5 x + 3) = - (x + 20)$

2. Solve the two equations for x

9 additional steps

$(5 x + 3) = (x + 20)$

Subtract from both sides:

$(5 x + 3) - x = (x + 20) - x$

Group like terms:

$(5 x - x) + 3 = (x + 20) - x$

Simplify the arithmetic:

$4 x + 3 = (x + 20) - x$

Group like terms:

$4 x + 3 = (x - x) + 20$

Simplify the arithmetic:

$4 x + 3 = 20$

Subtract from both sides:

$(4 x + 3) - 3 = 20 - 3$

Simplify the arithmetic:

$4 x = 20 - 3$

Simplify the arithmetic:

$4 x = 17$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{17}{4}$

Simplify the fraction:

$x = \frac{17}{4}$

10 additional steps

$(5 x + 3) = - (x + 20)$

Expand the parentheses:

$(5 x + 3) = - x - 20$

Add to both sides:

$(5 x + 3) + x = (- x - 20) + x$

Group like terms:

$(5 x + x) + 3 = (- x - 20) + x$

Simplify the arithmetic:

$6 x + 3 = (- x - 20) + x$

Group like terms:

$6 x + 3 = (- x + x) - 20$

Simplify the arithmetic:

$6 x + 3 = - 20$

Subtract from both sides:

$(6 x + 3) - 3 = - 20 - 3$

Simplify the arithmetic:

$6 x = - 20 - 3$

Simplify the arithmetic:

$6 x = - 23$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{- 23}{6}$

Simplify the fraction:

$x = \frac{- 23}{6}$

3. List the solutions

$x = \frac{17}{4}, - \frac{23}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 x + 3 |$
$y = | x + 20 |$
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