Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{22}{3}, \frac{16}{9}

x=\frac{22}{3} , \frac{16}{9}

Mixed number form:

x = 7 \frac{1}{3}, 1 \frac{7}{9}

x=7\frac{1}{3} , 1\frac{7}{9}

Decimal form:

x = 7.333, 1.778

x=7.333 , 1.778

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
$| 6 x - 19 | = | 3 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 x - 19 \| = \| 3 x + 3 \|$
$x = + y$	$(6 x - 19) = (3 x + 3)$
$x = - y$	$(6 x - 19) = - (3 x + 3)$
$+ x = y$	$(6 x - 19) = (3 x + 3)$
$- x = y$	$- (6 x - 19) = (3 x + 3)$

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:

$\| x \| = \| y \|$	$\| 6 x - 19 \| = \| 3 x + 3 \|$
$x = + y$ , $+ x = y$	$(6 x - 19) = (3 x + 3)$
$x = - y$ , $- x = y$	$(6 x - 19) = - (3 x + 3)$

2. Solve the two equations for x

9 additional steps

$(6 x - 19) = (3 x + 3)$

Subtract from both sides:

$(6 x - 19) - 3 x = (3 x + 3) - 3 x$

Group like terms:

$(6 x - 3 x) - 19 = (3 x + 3) - 3 x$

Simplify the arithmetic:

$3 x - 19 = (3 x + 3) - 3 x$

Group like terms:

$3 x - 19 = (3 x - 3 x) + 3$

Simplify the arithmetic:

$3 x - 19 = 3$

Add to both sides:

$(3 x - 19) + 19 = 3 + 19$

Simplify the arithmetic:

$3 x = 3 + 19$

Simplify the arithmetic:

$3 x = 22$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{22}{3}$

Simplify the fraction:

$x = \frac{22}{3}$

10 additional steps

$(6 x - 19) = - (3 x + 3)$

Expand the parentheses:

$(6 x - 19) = - 3 x - 3$

Add to both sides:

$(6 x - 19) + 3 x = (- 3 x - 3) + 3 x$

Group like terms:

$(6 x + 3 x) - 19 = (- 3 x - 3) + 3 x$

Simplify the arithmetic:

$9 x - 19 = (- 3 x - 3) + 3 x$

Group like terms:

$9 x - 19 = (- 3 x + 3 x) - 3$

Simplify the arithmetic:

$9 x - 19 = - 3$

Add to both sides:

$(9 x - 19) + 19 = - 3 + 19$

Simplify the arithmetic:

$9 x = - 3 + 19$

Simplify the arithmetic:

$9 x = 16$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{16}{9}$

Simplify the fraction:

$x = \frac{16}{9}$

3. List the solutions

$x = \frac{22}{3}, \frac{16}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 x - 19 |$
$y = | 3 x + 3 |$
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