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Solution - Absolute value equations

Exact form:

c = \frac{1}{2}, 9

c=\frac{1}{2} , 9

Decimal form:

c = 0.5, 9

c=0.5 , 9

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
$- | c + 8 | = - | - 3 c + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$- \| c + 8 \| = - \| - 3 c + 10 \|$
$x = + y$	$- (c + 8) = - (- 3 c + 10)$
$x = - y$	$- (c + 8) = - (- (- 3 c + 10))$
$+ x = y$	$- (c + 8) = - (- 3 c + 10)$
$- x = y$	$- (- (c + 8)) = - (- 3 c + 10)$

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:

$\| x \| = \| y \|$	$- \| c + 8 \| = - \| - 3 c + 10 \|$
$x = + y$ , $+ x = y$	$- (c + 8) = - (- 3 c + 10)$
$x = - y$ , $- x = y$	$- (c + 8) = - (- (- 3 c + 10))$

2. Solve the two equations for c

15 additional steps

$- (c + 8) = - (- 3 c + 10)$

Expand the parentheses:

$- c - 8 = - (- 3 c + 10)$

Expand the parentheses:

$- c - 8 = 3 c - 10$

Subtract from both sides:

$(- c - 8) - 3 c = (3 c - 10) - 3 c$

Group like terms:

$(- c - 3 c) - 8 = (3 c - 10) - 3 c$

Simplify the arithmetic:

$- 4 c - 8 = (3 c - 10) - 3 c$

Group like terms:

$- 4 c - 8 = (3 c - 3 c) - 10$

Simplify the arithmetic:

$- 4 c - 8 = - 10$

Add to both sides:

$(- 4 c - 8) + 8 = - 10 + 8$

Simplify the arithmetic:

$- 4 c = - 10 + 8$

Simplify the arithmetic:

$- 4 c = - 2$

Divide both sides by :

$\frac{(- 4 c)}{- 4} = \frac{- 2}{- 4}$

Cancel out the negatives:

$\frac{4 c}{4} = \frac{- 2}{- 4}$

Simplify the fraction:

$c = \frac{- 2}{- 4}$

Cancel out the negatives:

$c = \frac{2}{4}$

Find the greatest common factor of the numerator and denominator:

$c = \frac{(1 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

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

13 additional steps

$- (c + 8) = - (- (- 3 c + 10))$

Expand the parentheses:

$- c - 8 = - (- (- 3 c + 10))$

Resolve the double minus:

$- c - 8 = - 3 c + 10$

Add to both sides:

$(- c - 8) + 3 c = (- 3 c + 10) + 3 c$

Group like terms:

$(- c + 3 c) - 8 = (- 3 c + 10) + 3 c$

Simplify the arithmetic:

$2 c - 8 = (- 3 c + 10) + 3 c$

Group like terms:

$2 c - 8 = (- 3 c + 3 c) + 10$

Simplify the arithmetic:

$2 c - 8 = 10$

Add to both sides:

$(2 c - 8) + 8 = 10 + 8$

Simplify the arithmetic:

$2 c = 10 + 8$

Simplify the arithmetic:

$2 c = 18$

Divide both sides by :

$\frac{(2 c)}{2} = \frac{18}{2}$

Simplify the fraction:

$c = \frac{18}{2}$

Find the greatest common factor of the numerator and denominator:

$c = \frac{(9 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$c = 9$

3. List the solutions

$c = \frac{1}{2}, 9$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = - | c + 8 |$
$y = - | - 3 c + 10 |$
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 c

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 c

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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