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Solution - Absolute value equations

Exact form:

x = - \frac{3}{4}, \frac{3}{5}

x=-\frac{3}{4} , \frac{3}{5}

Decimal form:

x = - 0.75, 0.6

x=-0.75 , 0.6

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
$| x - 6 | = | 9 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 6 \| = \| 9 x \|$
$x = + y$	$(x - 6) = (9 x)$
$x = - y$	$(x - 6) = - (9 x)$
$+ x = y$	$(x - 6) = (9 x)$
$- x = y$	$- (x - 6) = (9 x)$

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:

$\| x \| = \| y \|$	$\| x - 6 \| = \| 9 x \|$
$x = + y$ , $+ x = y$	$(x - 6) = (9 x)$
$x = - y$ , $- x = y$	$(x - 6) = - (9 x)$

2. Solve the two equations for x

12 additional steps

$(x - 6) = 9 x$

Subtract from both sides:

$(x - 6) - 9 x = (9 x) - 9 x$

Group like terms:

$(x - 9 x) - 6 = (9 x) - 9 x$

Simplify the arithmetic:

$- 8 x - 6 = (9 x) - 9 x$

Simplify the arithmetic:

$- 8 x - 6 = 0$

Add to both sides:

$(- 8 x - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$- 8 x = 0 + 6$

Simplify the arithmetic:

$- 8 x = 6$

Divide both sides by :

$\frac{(- 8 x)}{- 8} = \frac{6}{- 8}$

Cancel out the negatives:

$\frac{8 x}{8} = \frac{6}{- 8}$

Simplify the fraction:

$x = \frac{6}{- 8}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 6}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 3 \cdot 2)}{(4 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 3}{4}$

9 additional steps

$(x - 6) = - 9 x$

Add to both sides:

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

Simplify the arithmetic:

$x = (- 9 x) + 6$

Add to both sides:

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

Simplify the arithmetic:

$10 x = ((- 9 x) + 6) + 9 x$

Group like terms:

$10 x = (- 9 x + 9 x) + 6$

Simplify the arithmetic:

$10 x = 6$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{6}{10}$

Simplify the fraction:

$x = \frac{6}{10}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 2)}{(5 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{3}{5}$

3. List the solutions

$x = - \frac{3}{4}, \frac{3}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 6 |$
$y = | 9 x |$
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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