Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{3}{4}, \frac{3}{14}

x=\frac{3}{4} , \frac{3}{14}

Decimal form:

x = 0.75, 0.214

x=0.75 , 0.214

See steps

Step-by-step explanation

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

$| 5 x | - | 9 x - 3 | = 0$

Add $| 9 x - 3 |$ to both sides of the equation:

$| 5 x | - | 9 x - 3 | + | 9 x - 3 | = | 9 x - 3 |$

Simplify the arithmetic

$| 5 x | = | 9 x - 3 |$

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
$| 5 x | = | 9 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x \| = \| 9 x - 3 \|$
$x = + y$	$(5 x) = (9 x - 3)$
$x = - y$	$(5 x) = (- (9 x - 3))$
$+ x = y$	$(5 x) = (9 x - 3)$
$- x = y$	$- (5 x) = (9 x - 3)$

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:

$\| x \| = \| y \|$	$\| 5 x \| = \| 9 x - 3 \|$
$x = + y$ , $+ x = y$	$(5 x) = (9 x - 3)$
$x = - y$ , $- x = y$	$(5 x) = (- (9 x - 3))$

3. Solve the two equations for x

7 additional steps

$5 x = (9 x - 3)$

Subtract from both sides:

$(5 x) - 9 x = (9 x - 3) - 9 x$

Simplify the arithmetic:

$- 4 x = (9 x - 3) - 9 x$

Group like terms:

$- 4 x = (9 x - 9 x) - 3$

Simplify the arithmetic:

$- 4 x = - 3$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 3}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 3}{- 4}$

Simplify the fraction:

$x = \frac{- 3}{- 4}$

Cancel out the negatives:

$x = \frac{3}{4}$

6 additional steps

$5 x = - (9 x - 3)$

Expand the parentheses:

$5 x = - 9 x + 3$

Add to both sides:

$(5 x) + 9 x = (- 9 x + 3) + 9 x$

Simplify the arithmetic:

$14 x = (- 9 x + 3) + 9 x$

Group like terms:

$14 x = (- 9 x + 9 x) + 3$

Simplify the arithmetic:

$14 x = 3$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{3}{14}$

Simplify the fraction:

$x = \frac{3}{14}$

4. List the solutions

$x = \frac{3}{4}, \frac{3}{14}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 5 x |$
$y = | 9 x - 3 |$
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 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