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Solution - Absolute value equations

Exact form:

x = - 1, - \frac{3}{7}

x=-1 , -\frac{3}{7}

Decimal form:

x = - 1, - 0.429

x=-1 , -0.429

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
$3 | 3 x + 1 | = 2 | 6 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| 3 x + 1 \| = 2 \| 6 x + 3 \|$
$x = + y$	$3 (3 x + 1) = 2 (6 x + 3)$
$x = - y$	$3 (3 x + 1) = 2 (- (6 x + 3))$
$+ x = y$	$3 (3 x + 1) = 2 (6 x + 3)$
$- x = y$	$3 (- (3 x + 1)) = 2 (6 x + 3)$

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:

$\| x \| = \| y \|$	$3 \| 3 x + 1 \| = 2 \| 6 x + 3 \|$
$x = + y$ , $+ x = y$	$3 (3 x + 1) = 2 (6 x + 3)$
$x = - y$ , $- x = y$	$3 (3 x + 1) = 2 (- (6 x + 3))$

2. Solve the two equations for x

18 additional steps

$3 \cdot (3 x + 1) = 2 \cdot (6 x + 3)$

Expand the parentheses:

$3 \cdot 3 x + 3 \cdot 1 = 2 \cdot (6 x + 3)$

Multiply the coefficients:

$9 x + 3 \cdot 1 = 2 \cdot (6 x + 3)$

Simplify the arithmetic:

$9 x + 3 = 2 \cdot (6 x + 3)$

Expand the parentheses:

$9 x + 3 = 2 \cdot 6 x + 2 \cdot 3$

Multiply the coefficients:

$9 x + 3 = 12 x + 2 \cdot 3$

Simplify the arithmetic:

$9 x + 3 = 12 x + 6$

Subtract from both sides:

$(9 x + 3) - 12 x = (12 x + 6) - 12 x$

Group like terms:

$(9 x - 12 x) + 3 = (12 x + 6) - 12 x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 3 x + 3 = 6$

Subtract from both sides:

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

Simplify the arithmetic:

$- 3 x = 6 - 3$

Simplify the arithmetic:

$- 3 x = 3$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{3}{- 3}$

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

Simplify the fraction:

$x = - 1$

18 additional steps

$3 \cdot (3 x + 1) = 2 \cdot (- (6 x + 3))$

Expand the parentheses:

$3 \cdot 3 x + 3 \cdot 1 = 2 \cdot (- (6 x + 3))$

Multiply the coefficients:

$9 x + 3 \cdot 1 = 2 \cdot (- (6 x + 3))$

Simplify the arithmetic:

$9 x + 3 = 2 \cdot (- (6 x + 3))$

Expand the parentheses:

$9 x + 3 = 2 \cdot (- 6 x - 3)$

Expand the parentheses:

$9 x + 3 = 2 \cdot - 6 x + 2 \cdot - 3$

Multiply the coefficients:

$9 x + 3 = - 12 x + 2 \cdot - 3$

Simplify the arithmetic:

$9 x + 3 = - 12 x - 6$

Add to both sides:

$(9 x + 3) + 12 x = (- 12 x - 6) + 12 x$

Group like terms:

$(9 x + 12 x) + 3 = (- 12 x - 6) + 12 x$

Simplify the arithmetic:

$21 x + 3 = (- 12 x - 6) + 12 x$

Group like terms:

$21 x + 3 = (- 12 x + 12 x) - 6$

Simplify the arithmetic:

$21 x + 3 = - 6$

Subtract from both sides:

$(21 x + 3) - 3 = - 6 - 3$

Simplify the arithmetic:

$21 x = - 6 - 3$

Simplify the arithmetic:

$21 x = - 9$

Divide both sides by :

$\frac{(21 x)}{21} = \frac{- 9}{21}$

Simplify the fraction:

$x = \frac{- 9}{21}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 3 \cdot 3)}{(7 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 3}{7}$

3. List the solutions

$x = - 1, - \frac{3}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | 3 x + 1 |$
$y = 2 | 6 x + 3 |$
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 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

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