Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{5}{2}, - \frac{1}{7}

x=\frac{5}{2} , -\frac{1}{7}

Mixed number form:

x = 2 \frac{1}{2}, - \frac{1}{7}

x=2\frac{1}{2} , -\frac{1}{7}

Decimal form:

x = 2.5, - 0.143

x=2.5 , -0.143

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
$| - 5 x - 6 | = | - 9 x + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 5 x - 6 \| = \| - 9 x + 4 \|$
$x = + y$	$(- 5 x - 6) = (- 9 x + 4)$
$x = - y$	$(- 5 x - 6) = - (- 9 x + 4)$
$+ x = y$	$(- 5 x - 6) = (- 9 x + 4)$
$- x = y$	$- (- 5 x - 6) = (- 9 x + 4)$

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:

$\| x \| = \| y \|$	$\| - 5 x - 6 \| = \| - 9 x + 4 \|$
$x = + y$ , $+ x = y$	$(- 5 x - 6) = (- 9 x + 4)$
$x = - y$ , $- x = y$	$(- 5 x - 6) = - (- 9 x + 4)$

2. Solve the two equations for x

11 additional steps

$(- 5 x - 6) = (- 9 x + 4)$

Add to both sides:

$(- 5 x - 6) + 9 x = (- 9 x + 4) + 9 x$

Group like terms:

$(- 5 x + 9 x) - 6 = (- 9 x + 4) + 9 x$

Simplify the arithmetic:

$4 x - 6 = (- 9 x + 4) + 9 x$

Group like terms:

$4 x - 6 = (- 9 x + 9 x) + 4$

Simplify the arithmetic:

$4 x - 6 = 4$

Add to both sides:

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

Simplify the arithmetic:

$4 x = 4 + 6$

Simplify the arithmetic:

$4 x = 10$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{10}{4}$

Simplify the fraction:

$x = \frac{10}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{5}{2}$

14 additional steps

$(- 5 x - 6) = - (- 9 x + 4)$

Expand the parentheses:

$(- 5 x - 6) = 9 x - 4$

Subtract from both sides:

$(- 5 x - 6) - 9 x = (9 x - 4) - 9 x$

Group like terms:

$(- 5 x - 9 x) - 6 = (9 x - 4) - 9 x$

Simplify the arithmetic:

$- 14 x - 6 = (9 x - 4) - 9 x$

Group like terms:

$- 14 x - 6 = (9 x - 9 x) - 4$

Simplify the arithmetic:

$- 14 x - 6 = - 4$

Add to both sides:

$(- 14 x - 6) + 6 = - 4 + 6$

Simplify the arithmetic:

$- 14 x = - 4 + 6$

Simplify the arithmetic:

$- 14 x = 2$

Divide both sides by :

$\frac{(- 14 x)}{- 14} = \frac{2}{- 14}$

Cancel out the negatives:

$\frac{14 x}{14} = \frac{2}{- 14}$

Simplify the fraction:

$x = \frac{2}{- 14}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 2}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{7}$

3. List the solutions

$x = \frac{5}{2}, - \frac{1}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 5 x - 6 |$
$y = | - 9 x + 4 |$
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