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 with one absolute value terms on each side

$| 9 x + 18 | - | 7 x + 14 | = 0$

Add $| 7 x + 14 |$ to both sides of the equation:

$| 9 x + 18 | - | 7 x + 14 | + | 7 x + 14 | = | 7 x + 14 |$

Simplify the arithmetic

$| 9 x + 18 | = | 7 x + 14 |$

2. 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
$| 9 x + 18 | = | 7 x + 14 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x + 18 \| = \| 7 x + 14 \|$
$x = + y$	$(9 x + 18) = (7 x + 14)$
$x = - y$	$(9 x + 18) = (- (7 x + 14))$
$+ x = y$	$(9 x + 18) = (7 x + 14)$
$- x = y$	$- (9 x + 18) = (7 x + 14)$

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:

$\| x \| = \| y \|$	$\| 9 x + 18 \| = \| 7 x + 14 \|$
$x = + y$ , $+ x = y$	$(9 x + 18) = (7 x + 14)$
$x = - y$ , $- x = y$	$(9 x + 18) = (- (7 x + 14))$

3. Solve the two equations for x

11 additional steps

$(9 x + 18) = (7 x + 14)$

Subtract from both sides:

$(9 x + 18) - 7 x = (7 x + 14) - 7 x$

Group like terms:

$(9 x - 7 x) + 18 = (7 x + 14) - 7 x$

Simplify the arithmetic:

$2 x + 18 = (7 x + 14) - 7 x$

Group like terms:

$2 x + 18 = (7 x - 7 x) + 14$

Simplify the arithmetic:

$2 x + 18 = 14$

Subtract from both sides:

$(2 x + 18) - 18 = 14 - 18$

Simplify the arithmetic:

$2 x = 14 - 18$

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:

$x = - 2$

12 additional steps

$(9 x + 18) = - (7 x + 14)$

Expand the parentheses:

$(9 x + 18) = - 7 x - 14$

Add to both sides:

$(9 x + 18) + 7 x = (- 7 x - 14) + 7 x$

Group like terms:

$(9 x + 7 x) + 18 = (- 7 x - 14) + 7 x$

Simplify the arithmetic:

$16 x + 18 = (- 7 x - 14) + 7 x$

Group like terms:

$16 x + 18 = (- 7 x + 7 x) - 14$

Simplify the arithmetic:

$16 x + 18 = - 14$

Subtract from both sides:

$(16 x + 18) - 18 = - 14 - 18$

Simplify the arithmetic:

$16 x = - 14 - 18$

Simplify the arithmetic:

$16 x = - 32$

Divide both sides by :

$\frac{(16 x)}{16} = \frac{- 32}{16}$

Simplify the fraction:

$x = \frac{- 32}{16}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 16)}{(1 \cdot 16)}$

Factor out and cancel the greatest common factor:

$x = - 2$

4. List the solutions

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

5. Graph

Each line represents the function of one side of the equation:
$y = | 9 x + 18 |$
$y = | 7 x + 14 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved