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Solution - Absolute value equations

Exact form:

f = - 1, \frac{1}{2}

f=-1 , \frac{1}{2}

Decimal form:

f = - 1, 0.5

f=-1 , 0.5

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
$| 3 f - 6 | = | 9 f |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 f - 6 \| = \| 9 f \|$
$x = + y$	$(3 f - 6) = (9 f)$
$x = - y$	$(3 f - 6) = - (9 f)$
$+ x = y$	$(3 f - 6) = (9 f)$
$- x = y$	$- (3 f - 6) = (9 f)$

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:

$\| x \| = \| y \|$	$\| 3 f - 6 \| = \| 9 f \|$
$x = + y$ , $+ x = y$	$(3 f - 6) = (9 f)$
$x = - y$ , $- x = y$	$(3 f - 6) = - (9 f)$

2. Solve the two equations for f

11 additional steps

$(3 f - 6) = 9 f$

Subtract from both sides:

$(3 f - 6) - 9 f = (9 f) - 9 f$

Group like terms:

$(3 f - 9 f) - 6 = (9 f) - 9 f$

Simplify the arithmetic:

$- 6 f - 6 = (9 f) - 9 f$

Simplify the arithmetic:

$- 6 f - 6 = 0$

Add to both sides:

$(- 6 f - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$- 6 f = 0 + 6$

Simplify the arithmetic:

$- 6 f = 6$

Divide both sides by :

$\frac{(- 6 f)}{- 6} = \frac{6}{- 6}$

Cancel out the negatives:

$\frac{6 f}{6} = \frac{6}{- 6}$

Simplify the fraction:

$f = \frac{6}{- 6}$

Move the negative sign from the denominator to the numerator:

$f = \frac{- 6}{6}$

Simplify the fraction:

$f = - 1$

9 additional steps

$(3 f - 6) = - 9 f$

Add to both sides:

$(3 f - 6) + 6 = (- 9 f) + 6$

Simplify the arithmetic:

$3 f = (- 9 f) + 6$

Add to both sides:

$(3 f) + 9 f = ((- 9 f) + 6) + 9 f$

Simplify the arithmetic:

$12 f = ((- 9 f) + 6) + 9 f$

Group like terms:

$12 f = (- 9 f + 9 f) + 6$

Simplify the arithmetic:

$12 f = 6$

Divide both sides by :

$\frac{(12 f)}{12} = \frac{6}{12}$

Simplify the fraction:

$f = \frac{6}{12}$

Find the greatest common factor of the numerator and denominator:

$f = \frac{(1 \cdot 6)}{(2 \cdot 6)}$

Factor out and cancel the greatest common factor:

$f = \frac{1}{2}$

3. List the solutions

$f = - 1, \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 f - 6 |$
$y = | 9 f |$
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 f

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 f

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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