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Solution - Absolute value equations

Exact form:

x = - 5, 3

x=-5 , 3

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
$| 3 x - 45 | = | 12 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x - 45 \| = \| 12 x \|$
$x = + y$	$(3 x - 45) = (12 x)$
$x = - y$	$(3 x - 45) = - (12 x)$
$+ x = y$	$(3 x - 45) = (12 x)$
$- x = y$	$- (3 x - 45) = (12 x)$

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:

$\| x \| = \| y \|$	$\| 3 x - 45 \| = \| 12 x \|$
$x = + y$ , $+ x = y$	$(3 x - 45) = (12 x)$
$x = - y$ , $- x = y$	$(3 x - 45) = - (12 x)$

2. Solve the two equations for x

12 additional steps

$(3 x - 45) = 12 x$

Subtract from both sides:

$(3 x - 45) - 12 x = (12 x) - 12 x$

Group like terms:

$(3 x - 12 x) - 45 = (12 x) - 12 x$

Simplify the arithmetic:

$- 9 x - 45 = (12 x) - 12 x$

Simplify the arithmetic:

$- 9 x - 45 = 0$

Add to both sides:

$(- 9 x - 45) + 45 = 0 + 45$

Simplify the arithmetic:

$- 9 x = 0 + 45$

Simplify the arithmetic:

$- 9 x = 45$

Divide both sides by :

$\frac{(- 9 x)}{- 9} = \frac{45}{- 9}$

Cancel out the negatives:

$\frac{9 x}{9} = \frac{45}{- 9}$

Simplify the fraction:

$x = \frac{45}{- 9}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 45}{9}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 9)}{(1 \cdot 9)}$

Factor out and cancel the greatest common factor:

$x = - 5$

9 additional steps

$(3 x - 45) = - 12 x$

Add to both sides:

$(3 x - 45) + 45 = (- 12 x) + 45$

Simplify the arithmetic:

$3 x = (- 12 x) + 45$

Add to both sides:

$(3 x) + 12 x = ((- 12 x) + 45) + 12 x$

Simplify the arithmetic:

$15 x = ((- 12 x) + 45) + 12 x$

Group like terms:

$15 x = (- 12 x + 12 x) + 45$

Simplify the arithmetic:

$15 x = 45$

Divide both sides by :

$\frac{(15 x)}{15} = \frac{45}{15}$

Simplify the fraction:

$x = \frac{45}{15}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 15)}{(1 \cdot 15)}$

Factor out and cancel the greatest common factor:

$x = 3$

3. List the solutions

$x = - 5, 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 x - 45 |$
$y = | 12 x |$
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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