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Solution - Absolute value equations

Exact form:

x = \frac{14}{5}, - \frac{10}{7}

x=\frac{14}{5} , -\frac{10}{7}

Mixed number form:

x = 2 \frac{4}{5}, - 1 \frac{3}{7}

x=2\frac{4}{5} , -1\frac{3}{7}

Decimal form:

x = 2.8, - 1.429

x=2.8 , -1.429

See steps

Step-by-step explanation

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

$| x + 12 | - 2 | 3 x - 1 | = 0$

Add $2 | 3 x - 1 |$ to both sides of the equation:

$| x + 12 | - 2 | 3 x - 1 | + 2 | 3 x - 1 | = 2 | 3 x - 1 |$

Simplify the arithmetic

$| x + 12 | = 2 | 3 x - 1 |$

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
$| x + 12 | = 2 | 3 x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x + 12 \| = 2 \| 3 x - 1 \|$
$x = + y$	$(x + 12) = 2 (3 x - 1)$
$x = - y$	$(x + 12) = 2 (- (3 x - 1))$
$+ x = y$	$(x + 12) = 2 (3 x - 1)$
$- x = y$	$- (x + 12) = 2 (3 x - 1)$

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:

$\| x \| = \| y \|$	$\| x + 12 \| = 2 \| 3 x - 1 \|$
$x = + y$ , $+ x = y$	$(x + 12) = 2 (3 x - 1)$
$x = - y$ , $- x = y$	$(x + 12) = 2 (- (3 x - 1))$

3. Solve the two equations for x

14 additional steps

$(x + 12) = 2 \cdot (3 x - 1)$

Expand the parentheses:

$(x + 12) = 2 \cdot 3 x + 2 \cdot - 1$

Multiply the coefficients:

$(x + 12) = 6 x + 2 \cdot - 1$

Simplify the arithmetic:

$(x + 12) = 6 x - 2$

Subtract from both sides:

$(x + 12) - 6 x = (6 x - 2) - 6 x$

Group like terms:

$(x - 6 x) + 12 = (6 x - 2) - 6 x$

Simplify the arithmetic:

$- 5 x + 12 = (6 x - 2) - 6 x$

Group like terms:

$- 5 x + 12 = (6 x - 6 x) - 2$

Simplify the arithmetic:

$- 5 x + 12 = - 2$

Subtract from both sides:

$(- 5 x + 12) - 12 = - 2 - 12$

Simplify the arithmetic:

$- 5 x = - 2 - 12$

Simplify the arithmetic:

$- 5 x = - 14$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{- 14}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{- 14}{- 5}$

Simplify the fraction:

$x = \frac{- 14}{- 5}$

Cancel out the negatives:

$x = \frac{14}{5}$

13 additional steps

$(x + 12) = 2 \cdot (- (3 x - 1))$

Expand the parentheses:

$(x + 12) = 2 \cdot (- 3 x + 1)$

Expand the parentheses:

$(x + 12) = 2 \cdot - 3 x + 2 \cdot 1$

Multiply the coefficients:

$(x + 12) = - 6 x + 2 \cdot 1$

Simplify the arithmetic:

$(x + 12) = - 6 x + 2$

Add to both sides:

$(x + 12) + 6 x = (- 6 x + 2) + 6 x$

Group like terms:

$(x + 6 x) + 12 = (- 6 x + 2) + 6 x$

Simplify the arithmetic:

$7 x + 12 = (- 6 x + 2) + 6 x$

Group like terms:

$7 x + 12 = (- 6 x + 6 x) + 2$

Simplify the arithmetic:

$7 x + 12 = 2$

Subtract from both sides:

$(7 x + 12) - 12 = 2 - 12$

Simplify the arithmetic:

$7 x = 2 - 12$

Simplify the arithmetic:

$7 x = - 10$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{- 10}{7}$

Simplify the fraction:

$x = \frac{- 10}{7}$

4. List the solutions

$x = \frac{14}{5}, - \frac{10}{7}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | x + 12 |$
$y = 2 | 3 x - 1 |$
The equation is true where the two lines cross.

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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