Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - 2, - 2

x=-2 , -2

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
$| 5 x + 10 | = - | 3 x + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x + 10 \| = - \| 3 x + 6 \|$
$x = + y$	$(5 x + 10) = - (3 x + 6)$
$x = - y$	$(5 x + 10) = - (- (3 x + 6))$
$+ x = y$	$(5 x + 10) = - (3 x + 6)$
$- x = y$	$- (5 x + 10) = - (3 x + 6)$

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:

$\| x \| = \| y \|$	$\| 5 x + 10 \| = - \| 3 x + 6 \|$
$x = + y$ , $+ x = y$	$(5 x + 10) = - (3 x + 6)$
$x = - y$ , $- x = y$	$(5 x + 10) = - (- (3 x + 6))$

2. Solve the two equations for x

12 additional steps

$(5 x + 10) = - (3 x + 6)$

Expand the parentheses:

$(5 x + 10) = - 3 x - 6$

Add to both sides:

$(5 x + 10) + 3 x = (- 3 x - 6) + 3 x$

Group like terms:

$(5 x + 3 x) + 10 = (- 3 x - 6) + 3 x$

Simplify the arithmetic:

$8 x + 10 = (- 3 x - 6) + 3 x$

Group like terms:

$8 x + 10 = (- 3 x + 3 x) - 6$

Simplify the arithmetic:

$8 x + 10 = - 6$

Subtract from both sides:

$(8 x + 10) - 10 = - 6 - 10$

Simplify the arithmetic:

$8 x = - 6 - 10$

Simplify the arithmetic:

$8 x = - 16$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{- 16}{8}$

Simplify the fraction:

$x = \frac{- 16}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 8)}{(1 \cdot 8)}$

Factor out and cancel the greatest common factor:

$x = - 2$

12 additional steps

$(5 x + 10) = - (- (3 x + 6))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(5 x + 10) = 3 x + 6$

Subtract from both sides:

$(5 x + 10) - 3 x = (3 x + 6) - 3 x$

Group like terms:

$(5 x - 3 x) + 10 = (3 x + 6) - 3 x$

Simplify the arithmetic:

$2 x + 10 = (3 x + 6) - 3 x$

Group like terms:

$2 x + 10 = (3 x - 3 x) + 6$

Simplify the arithmetic:

$2 x + 10 = 6$

Subtract from both sides:

$(2 x + 10) - 10 = 6 - 10$

Simplify the arithmetic:

$2 x = 6 - 10$

Simplify the arithmetic:

$2 x = - 4$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 4}{2}$

Simplify the fraction:

$x = \frac{- 4}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = - 2$

3. List the solutions

$x = - 2, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 x + 10 |$
$y = - | 3 x + 6 |$
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