Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{1}{3}

x=\frac{1}{3}

Decimal form:

x = 0.333

x=0.333

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
$4 | 3 x + 1 | = 12 | x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| 3 x + 1 \| = 12 \| x - 1 \|$
$x = + y$	$4 (3 x + 1) = 12 (x - 1)$
$x = - y$	$4 (3 x + 1) = 12 (- (x - 1))$
$+ x = y$	$4 (3 x + 1) = 12 (x - 1)$
$- x = y$	$4 (- (3 x + 1)) = 12 (x - 1)$

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:

$\| x \| = \| y \|$	$4 \| 3 x + 1 \| = 12 \| x - 1 \|$
$x = + y$ , $+ x = y$	$4 (3 x + 1) = 12 (x - 1)$
$x = - y$ , $- x = y$	$4 (3 x + 1) = 12 (- (x - 1))$

2. Solve the two equations for x

10 additional steps

$4 \cdot (3 x + 1) = 12 \cdot (x - 1)$

Expand the parentheses:

$4 \cdot 3 x + 4 \cdot 1 = 12 \cdot (x - 1)$

Multiply the coefficients:

$12 x + 4 \cdot 1 = 12 \cdot (x - 1)$

Simplify the arithmetic:

$12 x + 4 = 12 \cdot (x - 1)$

Expand the parentheses:

$12 x + 4 = 12 x + 12 \cdot - 1$

Simplify the arithmetic:

$12 x + 4 = 12 x - 12$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 = - 12$

The statement is false:

$4 = - 12$

The equation is false so it has no solution.

19 additional steps

$4 \cdot (3 x + 1) = 12 \cdot (- (x - 1))$

Expand the parentheses:

$4 \cdot 3 x + 4 \cdot 1 = 12 \cdot (- (x - 1))$

Multiply the coefficients:

$12 x + 4 \cdot 1 = 12 \cdot (- (x - 1))$

Simplify the arithmetic:

$12 x + 4 = 12 \cdot (- (x - 1))$

Expand the parentheses:

$12 x + 4 = 12 \cdot (- x + 1)$

$12 x + 4 = 12 \cdot - x + 12 \cdot 1$

Group like terms:

$12 x + 4 = (12 \cdot - 1) x + 12 \cdot 1$

Multiply the coefficients:

$12 x + 4 = - 12 x + 12 \cdot 1$

Simplify the arithmetic:

$12 x + 4 = - 12 x + 12$

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$24 x + 4 = (- 12 x + 12) + 12 x$

Group like terms:

$24 x + 4 = (- 12 x + 12 x) + 12$

Simplify the arithmetic:

$24 x + 4 = 12$

Subtract from both sides:

$(24 x + 4) - 4 = 12 - 4$

Simplify the arithmetic:

$24 x = 12 - 4$

Simplify the arithmetic:

$24 x = 8$

Divide both sides by :

$\frac{(24 x)}{24} = \frac{8}{24}$

Simplify the fraction:

$x = \frac{8}{24}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 8)}{(3 \cdot 8)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{3}$

3. Graph

Each line represents the function of one side of the equation:
$y = 4 | 3 x + 1 |$
$y = 12 | x - 1 |$
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 without absolute value bars

2. Solve the two equations for x

3. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved