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Solution - Absolute value equations

Exact form:

x = 8, \frac{22}{9}

x=8 , \frac{22}{9}

Mixed number form:

x = 8, 2 \frac{4}{9}

x=8 , 2\frac{4}{9}

Decimal form:

x = 8, 2.444

x=8 , 2.444

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
$| 3 x + 1 | = | 6 x - 23 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x + 1 \| = \| 6 x - 23 \|$
$x = + y$	$(3 x + 1) = (6 x - 23)$
$x = - y$	$(3 x + 1) = - (6 x - 23)$
$+ x = y$	$(3 x + 1) = (6 x - 23)$
$- x = y$	$- (3 x + 1) = (6 x - 23)$

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:

$\| x \| = \| y \|$	$\| 3 x + 1 \| = \| 6 x - 23 \|$
$x = + y$ , $+ x = y$	$(3 x + 1) = (6 x - 23)$
$x = - y$ , $- x = y$	$(3 x + 1) = - (6 x - 23)$

2. Solve the two equations for x

13 additional steps

$(3 x + 1) = (6 x - 23)$

Subtract from both sides:

$(3 x + 1) - 6 x = (6 x - 23) - 6 x$

Group like terms:

$(3 x - 6 x) + 1 = (6 x - 23) - 6 x$

Simplify the arithmetic:

$- 3 x + 1 = (6 x - 23) - 6 x$

Group like terms:

$- 3 x + 1 = (6 x - 6 x) - 23$

Simplify the arithmetic:

$- 3 x + 1 = - 23$

Subtract from both sides:

$(- 3 x + 1) - 1 = - 23 - 1$

Simplify the arithmetic:

$- 3 x = - 23 - 1$

Simplify the arithmetic:

$- 3 x = - 24$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{- 24}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{- 24}{- 3}$

Simplify the fraction:

$x = \frac{- 24}{- 3}$

Cancel out the negatives:

$x = \frac{24}{3}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = 8$

10 additional steps

$(3 x + 1) = - (6 x - 23)$

Expand the parentheses:

$(3 x + 1) = - 6 x + 23$

Add to both sides:

$(3 x + 1) + 6 x = (- 6 x + 23) + 6 x$

Group like terms:

$(3 x + 6 x) + 1 = (- 6 x + 23) + 6 x$

Simplify the arithmetic:

$9 x + 1 = (- 6 x + 23) + 6 x$

Group like terms:

$9 x + 1 = (- 6 x + 6 x) + 23$

Simplify the arithmetic:

$9 x + 1 = 23$

Subtract from both sides:

$(9 x + 1) - 1 = 23 - 1$

Simplify the arithmetic:

$9 x = 23 - 1$

Simplify the arithmetic:

$9 x = 22$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{22}{9}$

Simplify the fraction:

$x = \frac{22}{9}$

3. List the solutions

$x = 8, \frac{22}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 x + 1 |$
$y = | 6 x - 23 |$
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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