Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - 4

x=-4

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
$| 2 x + 6 | = 2 | x + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x + 6 \| = 2 \| x + 5 \|$
$x = + y$	$(2 x + 6) = 2 (x + 5)$
$x = - y$	$(2 x + 6) = 2 (- (x + 5))$
$+ x = y$	$(2 x + 6) = 2 (x + 5)$
$- x = y$	$- (2 x + 6) = 2 (x + 5)$

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:

$\| x \| = \| y \|$	$\| 2 x + 6 \| = 2 \| x + 5 \|$
$x = + y$ , $+ x = y$	$(2 x + 6) = 2 (x + 5)$
$x = - y$ , $- x = y$	$(2 x + 6) = 2 (- (x + 5))$

2. Solve the two equations for x

7 additional steps

$(2 x + 6) = 2 \cdot (x + 5)$

Expand the parentheses:

$(2 x + 6) = 2 x + 2 \cdot 5$

Simplify the arithmetic:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$6 = 10$

The statement is false:

$6 = 10$

The equation is false so it has no solution.

16 additional steps

$(2 x + 6) = 2 \cdot (- (x + 5))$

Expand the parentheses:

$(2 x + 6) = 2 \cdot (- x - 5)$

$(2 x + 6) = 2 \cdot - x + 2 \cdot - 5$

Group like terms:

$(2 x + 6) = (2 \cdot - 1) x + 2 \cdot - 5$

Multiply the coefficients:

$(2 x + 6) = - 2 x + 2 \cdot - 5$

Simplify the arithmetic:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$4 x + 6 = (- 2 x - 10) + 2 x$

Group like terms:

$4 x + 6 = (- 2 x + 2 x) - 10$

Simplify the arithmetic:

$4 x + 6 = - 10$

Subtract from both sides:

$(4 x + 6) - 6 = - 10 - 6$

Simplify the arithmetic:

$4 x = - 10 - 6$

Simplify the arithmetic:

$4 x = - 16$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{- 16}{4}$

Simplify the fraction:

$x = \frac{- 16}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 4 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = - 4$

3. Graph

Each line represents the function of one side of the equation:
$y = | 2 x + 6 |$
$y = 2 | x + 5 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved