Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = 2, \frac{10}{9}

x=2 , \frac{10}{9}

Mixed number form:

x = 2, 1 \frac{1}{9}

x=2 , 1\frac{1}{9}

Decimal form:

x = 2, 1.111

x=2 , 1.111

See steps

Step-by-step explanation

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

$4 | 4 x - 4 | - 4 | 5 x - 6 | = 0$

Add $4 | 5 x - 6 |$ to both sides of the equation:

$4 | 4 x - 4 | - 4 | 5 x - 6 | + 4 | 5 x - 6 | = 4 | 5 x - 6 |$

Simplify the arithmetic

$4 | 4 x - 4 | = 4 | 5 x - 6 |$

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

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

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:

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

3. Solve the two equations for x

19 additional steps

$4 \cdot (4 x - 4) = 4 \cdot (5 x - 6)$

Expand the parentheses:

$4 \cdot 4 x + 4 \cdot - 4 = 4 \cdot (5 x - 6)$

Multiply the coefficients:

$16 x + 4 \cdot - 4 = 4 \cdot (5 x - 6)$

Simplify the arithmetic:

$16 x - 16 = 4 \cdot (5 x - 6)$

Expand the parentheses:

$16 x - 16 = 4 \cdot 5 x + 4 \cdot - 6$

Multiply the coefficients:

$16 x - 16 = 20 x + 4 \cdot - 6$

Simplify the arithmetic:

$16 x - 16 = 20 x - 24$

Subtract from both sides:

$(16 x - 16) - 20 x = (20 x - 24) - 20 x$

Group like terms:

$(16 x - 20 x) - 16 = (20 x - 24) - 20 x$

Simplify the arithmetic:

$- 4 x - 16 = (20 x - 24) - 20 x$

Group like terms:

$- 4 x - 16 = (20 x - 20 x) - 24$

Simplify the arithmetic:

$- 4 x - 16 = - 24$

Add to both sides:

$(- 4 x - 16) + 16 = - 24 + 16$

Simplify the arithmetic:

$- 4 x = - 24 + 16$

Simplify the arithmetic:

$- 4 x = - 8$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 8}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 8}{- 4}$

Simplify the fraction:

$x = \frac{- 8}{- 4}$

Cancel out the negatives:

$x = \frac{8}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = 2$

18 additional steps

$4 \cdot (4 x - 4) = 4 \cdot (- (5 x - 6))$

Expand the parentheses:

$4 \cdot 4 x + 4 \cdot - 4 = 4 \cdot (- (5 x - 6))$

Multiply the coefficients:

$16 x + 4 \cdot - 4 = 4 \cdot (- (5 x - 6))$

Simplify the arithmetic:

$16 x - 16 = 4 \cdot (- (5 x - 6))$

Expand the parentheses:

$16 x - 16 = 4 \cdot (- 5 x + 6)$

Expand the parentheses:

$16 x - 16 = 4 \cdot - 5 x + 4 \cdot 6$

Multiply the coefficients:

$16 x - 16 = - 20 x + 4 \cdot 6$

Simplify the arithmetic:

$16 x - 16 = - 20 x + 24$

Add to both sides:

$(16 x - 16) + 20 x = (- 20 x + 24) + 20 x$

Group like terms:

$(16 x + 20 x) - 16 = (- 20 x + 24) + 20 x$

Simplify the arithmetic:

$36 x - 16 = (- 20 x + 24) + 20 x$

Group like terms:

$36 x - 16 = (- 20 x + 20 x) + 24$

Simplify the arithmetic:

$36 x - 16 = 24$

Add to both sides:

$(36 x - 16) + 16 = 24 + 16$

Simplify the arithmetic:

$36 x = 24 + 16$

Simplify the arithmetic:

$36 x = 40$

Divide both sides by :

$\frac{(36 x)}{36} = \frac{40}{36}$

Simplify the fraction:

$x = \frac{40}{36}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(10 \cdot 4)}{(9 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{10}{9}$

4. List the solutions

$x = 2, \frac{10}{9}$
(2 solution(s))

5. Graph

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