Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{3}{4}, \frac{1}{4}

x=\frac{3}{4} , \frac{1}{4}

Decimal form:

x = 0.75, 0.25

x=0.75 , 0.25

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
$| 8 x - 3 | = | 4 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 x - 3 \| = \| 4 x \|$
$x = + y$	$(8 x - 3) = (4 x)$
$x = - y$	$(8 x - 3) = - (4 x)$
$+ x = y$	$(8 x - 3) = (4 x)$
$- x = y$	$- (8 x - 3) = (4 x)$

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:

$\| x \| = \| y \|$	$\| 8 x - 3 \| = \| 4 x \|$
$x = + y$ , $+ x = y$	$(8 x - 3) = (4 x)$
$x = - y$ , $- x = y$	$(8 x - 3) = - (4 x)$

2. Solve the two equations for x

8 additional steps

$(8 x - 3) = 4 x$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$4 x - 3 = 0$

Add to both sides:

$(4 x - 3) + 3 = 0 + 3$

Simplify the arithmetic:

$4 x = 0 + 3$

Simplify the arithmetic:

$4 x = 3$

Divide both sides by :

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

Simplify the fraction:

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

9 additional steps

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

Add to both sides:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

$12 x = ((- 4 x) + 3) + 4 x$

Group like terms:

$12 x = (- 4 x + 4 x) + 3$

Simplify the arithmetic:

$12 x = 3$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{3}{12}$

Simplify the fraction:

$x = \frac{3}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 3)}{(4 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{4}$

3. List the solutions

$x = \frac{3}{4}, \frac{1}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 8 x - 3 |$
$y = | 4 x |$
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