Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{53}{4}, \frac{59}{8}

x=\frac{53}{4} , \frac{59}{8}

Mixed number form:

x = 13 \frac{1}{4}, 7 \frac{3}{8}

x=13\frac{1}{4} , 7\frac{3}{8}

Decimal form:

x = 13.25, 7.375

x=13.25 , 7.375

See steps

Step-by-step explanation

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

$| 6 x - 56 | - | 2 x - 3 | = 0$

Add $| 2 x - 3 |$ to both sides of the equation:

$| 6 x - 56 | - | 2 x - 3 | + | 2 x - 3 | = | 2 x - 3 |$

Simplify the arithmetic

$| 6 x - 56 | = | 2 x - 3 |$

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
$| 6 x - 56 | = | 2 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 x - 56 \| = \| 2 x - 3 \|$
$x = + y$	$(6 x - 56) = (2 x - 3)$
$x = - y$	$(6 x - 56) = (- (2 x - 3))$
$+ x = y$	$(6 x - 56) = (2 x - 3)$
$- x = y$	$- (6 x - 56) = (2 x - 3)$

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:

$\| x \| = \| y \|$	$\| 6 x - 56 \| = \| 2 x - 3 \|$
$x = + y$ , $+ x = y$	$(6 x - 56) = (2 x - 3)$
$x = - y$ , $- x = y$	$(6 x - 56) = (- (2 x - 3))$

3. Solve the two equations for x

9 additional steps

$(6 x - 56) = (2 x - 3)$

Subtract from both sides:

$(6 x - 56) - 2 x = (2 x - 3) - 2 x$

Group like terms:

$(6 x - 2 x) - 56 = (2 x - 3) - 2 x$

Simplify the arithmetic:

$4 x - 56 = (2 x - 3) - 2 x$

Group like terms:

$4 x - 56 = (2 x - 2 x) - 3$

Simplify the arithmetic:

$4 x - 56 = - 3$

Add to both sides:

$(4 x - 56) + 56 = - 3 + 56$

Simplify the arithmetic:

$4 x = - 3 + 56$

Simplify the arithmetic:

$4 x = 53$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{53}{4}$

Simplify the fraction:

$x = \frac{53}{4}$

10 additional steps

$(6 x - 56) = - (2 x - 3)$

Expand the parentheses:

$(6 x - 56) = - 2 x + 3$

Add to both sides:

$(6 x - 56) + 2 x = (- 2 x + 3) + 2 x$

Group like terms:

$(6 x + 2 x) - 56 = (- 2 x + 3) + 2 x$

Simplify the arithmetic:

$8 x - 56 = (- 2 x + 3) + 2 x$

Group like terms:

$8 x - 56 = (- 2 x + 2 x) + 3$

Simplify the arithmetic:

$8 x - 56 = 3$

Add to both sides:

$(8 x - 56) + 56 = 3 + 56$

Simplify the arithmetic:

$8 x = 3 + 56$

Simplify the arithmetic:

$8 x = 59$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{59}{8}$

Simplify the fraction:

$x = \frac{59}{8}$

4. List the solutions

$x = \frac{53}{4}, \frac{59}{8}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 6 x - 56 |$
$y = | 2 x - 3 |$
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