Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{7}{5}, - 1

x=\frac{7}{5} , -1

Mixed number form:

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

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

Decimal form:

x = 1.4, - 1

x=1.4 , -1

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
$2 | 4 x + 1 | = 3 | x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| 4 x + 1 \| = 3 \| x + 3 \|$
$x = + y$	$2 (4 x + 1) = 3 (x + 3)$
$x = - y$	$2 (4 x + 1) = 3 (- (x + 3))$
$+ x = y$	$2 (4 x + 1) = 3 (x + 3)$
$- x = y$	$2 (- (4 x + 1)) = 3 (x + 3)$

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:

$\| x \| = \| y \|$	$2 \| 4 x + 1 \| = 3 \| x + 3 \|$
$x = + y$ , $+ x = y$	$2 (4 x + 1) = 3 (x + 3)$
$x = - y$ , $- x = y$	$2 (4 x + 1) = 3 (- (x + 3))$

2. Solve the two equations for x

14 additional steps

$2 \cdot (4 x + 1) = 3 \cdot (x + 3)$

Expand the parentheses:

$2 \cdot 4 x + 2 \cdot 1 = 3 \cdot (x + 3)$

Multiply the coefficients:

$8 x + 2 \cdot 1 = 3 \cdot (x + 3)$

Simplify the arithmetic:

$8 x + 2 = 3 \cdot (x + 3)$

Expand the parentheses:

$8 x + 2 = 3 x + 3 \cdot 3$

Simplify the arithmetic:

$8 x + 2 = 3 x + 9$

Subtract from both sides:

$(8 x + 2) - 3 x = (3 x + 9) - 3 x$

Group like terms:

$(8 x - 3 x) + 2 = (3 x + 9) - 3 x$

Simplify the arithmetic:

$5 x + 2 = (3 x + 9) - 3 x$

Group like terms:

$5 x + 2 = (3 x - 3 x) + 9$

Simplify the arithmetic:

$5 x + 2 = 9$

Subtract from both sides:

$(5 x + 2) - 2 = 9 - 2$

Simplify the arithmetic:

$5 x = 9 - 2$

Simplify the arithmetic:

$5 x = 7$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{7}{5}$

Simplify the fraction:

$x = \frac{7}{5}$

18 additional steps

$2 \cdot (4 x + 1) = 3 \cdot (- (x + 3))$

Expand the parentheses:

$2 \cdot 4 x + 2 \cdot 1 = 3 \cdot (- (x + 3))$

Multiply the coefficients:

$8 x + 2 \cdot 1 = 3 \cdot (- (x + 3))$

Simplify the arithmetic:

$8 x + 2 = 3 \cdot (- (x + 3))$

Expand the parentheses:

$8 x + 2 = 3 \cdot (- x - 3)$

$8 x + 2 = 3 \cdot - x + 3 \cdot - 3$

Group like terms:

$8 x + 2 = (3 \cdot - 1) x + 3 \cdot - 3$

Multiply the coefficients:

$8 x + 2 = - 3 x + 3 \cdot - 3$

Simplify the arithmetic:

$8 x + 2 = - 3 x - 9$

Add to both sides:

$(8 x + 2) + 3 x = (- 3 x - 9) + 3 x$

Group like terms:

$(8 x + 3 x) + 2 = (- 3 x - 9) + 3 x$

Simplify the arithmetic:

$11 x + 2 = (- 3 x - 9) + 3 x$

Group like terms:

$11 x + 2 = (- 3 x + 3 x) - 9$

Simplify the arithmetic:

$11 x + 2 = - 9$

Subtract from both sides:

$(11 x + 2) - 2 = - 9 - 2$

Simplify the arithmetic:

$11 x = - 9 - 2$

Simplify the arithmetic:

$11 x = - 11$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{- 11}{11}$

Simplify the fraction:

$x = \frac{- 11}{11}$

Simplify the fraction:

$x = - 1$

3. List the solutions

$x = \frac{7}{5}, - 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | 4 x + 1 |$
$y = 3 | x + 3 |$
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 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