Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{37}{59}, \frac{35}{61}

x=\frac{37}{59} , \frac{35}{61}

Decimal form:

x = 0.627, 0.574

x=0.627 , 0.574

See steps

Step-by-step explanation

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

$12 | 5 x - 3 | - | x + 1 | = 0$

Add $| x + 1 |$ to both sides of the equation:

$12 | 5 x - 3 | - | x + 1 | + | x + 1 | = | x + 1 |$

Simplify the arithmetic

$12 | 5 x - 3 | = | x + 1 |$

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
$12 | 5 x - 3 | = | x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$12 \| 5 x - 3 \| = \| x + 1 \|$
$x = + y$	$12 (5 x - 3) = (x + 1)$
$x = - y$	$12 (5 x - 3) = (- (x + 1))$
$+ x = y$	$12 (5 x - 3) = (x + 1)$
$- x = y$	$12 (- (5 x - 3)) = (x + 1)$

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:

$\| x \| = \| y \|$	$12 \| 5 x - 3 \| = \| x + 1 \|$
$x = + y$ , $+ x = y$	$12 (5 x - 3) = (x + 1)$
$x = - y$ , $- x = y$	$12 (5 x - 3) = (- (x + 1))$

3. Solve the two equations for x

12 additional steps

$12 \cdot (5 x - 3) = (x + 1)$

Expand the parentheses:

$12 \cdot 5 x + 12 \cdot - 3 = (x + 1)$

Multiply the coefficients:

$60 x + 12 \cdot - 3 = (x + 1)$

Simplify the arithmetic:

$60 x - 36 = (x + 1)$

Subtract from both sides:

$(60 x - 36) - x = (x + 1) - x$

Group like terms:

$(60 x - x) - 36 = (x + 1) - x$

Simplify the arithmetic:

$59 x - 36 = (x + 1) - x$

Group like terms:

$59 x - 36 = (x - x) + 1$

Simplify the arithmetic:

$59 x - 36 = 1$

Add to both sides:

$(59 x - 36) + 36 = 1 + 36$

Simplify the arithmetic:

$59 x = 1 + 36$

Simplify the arithmetic:

$59 x = 37$

Divide both sides by :

$\frac{(59 x)}{59} = \frac{37}{59}$

Simplify the fraction:

$x = \frac{37}{59}$

13 additional steps

$12 \cdot (5 x - 3) = (- (x + 1))$

Expand the parentheses:

$12 \cdot 5 x + 12 \cdot - 3 = (- (x + 1))$

Multiply the coefficients:

$60 x + 12 \cdot - 3 = (- (x + 1))$

Simplify the arithmetic:

$60 x - 36 = (- (x + 1))$

Expand the parentheses:

$60 x - 36 = - x - 1$

Add to both sides:

$(60 x - 36) + x = (- x - 1) + x$

Group like terms:

$(60 x + x) - 36 = (- x - 1) + x$

Simplify the arithmetic:

$61 x - 36 = (- x - 1) + x$

Group like terms:

$61 x - 36 = (- x + x) - 1$

Simplify the arithmetic:

$61 x - 36 = - 1$

Add to both sides:

$(61 x - 36) + 36 = - 1 + 36$

Simplify the arithmetic:

$61 x = - 1 + 36$

Simplify the arithmetic:

$61 x = 35$

Divide both sides by :

$\frac{(61 x)}{61} = \frac{35}{61}$

Simplify the fraction:

$x = \frac{35}{61}$

4. List the solutions

$x = \frac{37}{59}, \frac{35}{61}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 12 | 5 x - 3 |$
$y = | x + 1 |$
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