Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{12}{5}, \frac{4}{3}

x=\frac{12}{5} , \frac{4}{3}

Mixed number form:

x = 2 \frac{2}{5}, 1 \frac{1}{3}

x=2\frac{2}{5} , 1\frac{1}{3}

Decimal form:

x = 2.4, 1.333

x=2.4 , 1.333

See steps

Step-by-step explanation

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

$| 4 x - 8 | + | x - 4 | = 0$

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

$| 4 x - 8 | + | x - 4 | - | x - 4 | = - | x - 4 |$

Simplify the arithmetic

$| 4 x - 8 | = - | x - 4 |$

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
$| 4 x - 8 | = - | x - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 8 \| = - \| x - 4 \|$
$x = + y$	$(4 x - 8) = - (x - 4)$
$x = - y$	$(4 x - 8) = - - (x - 4)$
$+ x = y$	$(4 x - 8) = - (x - 4)$
$- x = y$	$- (4 x - 8) = - (x - 4)$

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:

$\| x \| = \| y \|$	$\| 4 x - 8 \| = - \| x - 4 \|$
$x = + y$ , $+ x = y$	$(4 x - 8) = - (x - 4)$
$x = - y$ , $- x = y$	$(4 x - 8) = - - (x - 4)$

3. Solve the two equations for x

10 additional steps

$(4 x - 8) = - (x - 4)$

Expand the parentheses:

$(4 x - 8) = - x + 4$

Add to both sides:

$(4 x - 8) + x = (- x + 4) + x$

Group like terms:

$(4 x + x) - 8 = (- x + 4) + x$

Simplify the arithmetic:

$5 x - 8 = (- x + 4) + x$

Group like terms:

$5 x - 8 = (- x + x) + 4$

Simplify the arithmetic:

$5 x - 8 = 4$

Add to both sides:

$(5 x - 8) + 8 = 4 + 8$

Simplify the arithmetic:

$5 x = 4 + 8$

Simplify the arithmetic:

$5 x = 12$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{12}{5}$

Simplify the fraction:

$x = \frac{12}{5}$

10 additional steps

$(4 x - 8) = - (- (x - 4))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(4 x - 8) = x - 4$

Subtract from both sides:

$(4 x - 8) - x = (x - 4) - x$

Group like terms:

$(4 x - x) - 8 = (x - 4) - x$

Simplify the arithmetic:

$3 x - 8 = (x - 4) - x$

Group like terms:

$3 x - 8 = (x - x) - 4$

Simplify the arithmetic:

$3 x - 8 = - 4$

Add to both sides:

$(3 x - 8) + 8 = - 4 + 8$

Simplify the arithmetic:

$3 x = - 4 + 8$

Simplify the arithmetic:

$3 x = 4$

Divide both sides by :

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

Simplify the fraction:

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

4. List the solutions

$x = \frac{12}{5}, \frac{4}{3}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 8 |$
$y = - | x - 4 |$
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 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