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Solution - Absolute value equations

Exact form:

x = - 6, - \frac{18}{5}

x=-6 , -\frac{18}{5}

Mixed number form:

x = - 6, - 3 \frac{3}{5}

x=-6 , -3\frac{3}{5}

Decimal form:

x = - 6, - 3.6

x=-6 , -3.6

See steps

Step-by-step explanation

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

$| 2 x + 6 | - | 3 x + 12 | = 0$

Add $| 3 x + 12 |$ to both sides of the equation:

$| 2 x + 6 | - | 3 x + 12 | + | 3 x + 12 | = | 3 x + 12 |$

Simplify the arithmetic

$| 2 x + 6 | = | 3 x + 12 |$

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
$| 2 x + 6 | = | 3 x + 12 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x + 6 \| = \| 3 x + 12 \|$
$x = + y$	$(2 x + 6) = (3 x + 12)$
$x = - y$	$(2 x + 6) = (- (3 x + 12))$
$+ x = y$	$(2 x + 6) = (3 x + 12)$
$- x = y$	$- (2 x + 6) = (3 x + 12)$

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:

$\| x \| = \| y \|$	$\| 2 x + 6 \| = \| 3 x + 12 \|$
$x = + y$ , $+ x = y$	$(2 x + 6) = (3 x + 12)$
$x = - y$ , $- x = y$	$(2 x + 6) = (- (3 x + 12))$

3. Solve the two equations for x

10 additional steps

$(2 x + 6) = (3 x + 12)$

Subtract from both sides:

$(2 x + 6) - 3 x = (3 x + 12) - 3 x$

Group like terms:

$(2 x - 3 x) + 6 = (3 x + 12) - 3 x$

Simplify the arithmetic:

$- x + 6 = (3 x + 12) - 3 x$

Group like terms:

$- x + 6 = (3 x - 3 x) + 12$

Simplify the arithmetic:

$- x + 6 = 12$

Subtract from both sides:

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

Simplify the arithmetic:

$- x = 12 - 6$

Simplify the arithmetic:

$- x = 6$

Multiply both sides by :

$- x \cdot - 1 = 6 \cdot - 1$

Remove the one(s):

$x = 6 \cdot - 1$

Simplify the arithmetic:

$x = - 6$

10 additional steps

$(2 x + 6) = - (3 x + 12)$

Expand the parentheses:

$(2 x + 6) = - 3 x - 12$

Add to both sides:

$(2 x + 6) + 3 x = (- 3 x - 12) + 3 x$

Group like terms:

$(2 x + 3 x) + 6 = (- 3 x - 12) + 3 x$

Simplify the arithmetic:

$5 x + 6 = (- 3 x - 12) + 3 x$

Group like terms:

$5 x + 6 = (- 3 x + 3 x) - 12$

Simplify the arithmetic:

$5 x + 6 = - 12$

Subtract from both sides:

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

Simplify the arithmetic:

$5 x = - 12 - 6$

Simplify the arithmetic:

$5 x = - 18$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{- 18}{5}$

Simplify the fraction:

$x = \frac{- 18}{5}$

4. List the solutions

$x = - 6, - \frac{18}{5}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 2 x + 6 |$
$y = | 3 x + 12 |$
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