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Solution - Absolute value equations

Exact form:

c = - 1, - \frac{5}{2}

c=-1 , -\frac{5}{2}

Mixed number form:

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

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

Decimal form:

c = - 1, - 2.5

c=-1 , -2.5

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
$| 3 c + 6 | = | c + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 c + 6 \| = \| c + 4 \|$
$x = + y$	$(3 c + 6) = (c + 4)$
$x = - y$	$(3 c + 6) = - (c + 4)$
$+ x = y$	$(3 c + 6) = (c + 4)$
$- x = y$	$- (3 c + 6) = (c + 4)$

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:

$\| x \| = \| y \|$	$\| 3 c + 6 \| = \| c + 4 \|$
$x = + y$ , $+ x = y$	$(3 c + 6) = (c + 4)$
$x = - y$ , $- x = y$	$(3 c + 6) = - (c + 4)$

2. Solve the two equations for c

10 additional steps

$(3 c + 6) = (c + 4)$

Subtract from both sides:

$(3 c + 6) - c = (c + 4) - c$

Group like terms:

$(3 c - c) + 6 = (c + 4) - c$

Simplify the arithmetic:

$2 c + 6 = (c + 4) - c$

Group like terms:

$2 c + 6 = (c - c) + 4$

Simplify the arithmetic:

$2 c + 6 = 4$

Subtract from both sides:

$(2 c + 6) - 6 = 4 - 6$

Simplify the arithmetic:

$2 c = 4 - 6$

Simplify the arithmetic:

$2 c = - 2$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$c = - 1$

12 additional steps

$(3 c + 6) = - (c + 4)$

Expand the parentheses:

$(3 c + 6) = - c - 4$

Add to both sides:

$(3 c + 6) + c = (- c - 4) + c$

Group like terms:

$(3 c + c) + 6 = (- c - 4) + c$

Simplify the arithmetic:

$4 c + 6 = (- c - 4) + c$

Group like terms:

$4 c + 6 = (- c + c) - 4$

Simplify the arithmetic:

$4 c + 6 = - 4$

Subtract from both sides:

$(4 c + 6) - 6 = - 4 - 6$

Simplify the arithmetic:

$4 c = - 4 - 6$

Simplify the arithmetic:

$4 c = - 10$

Divide both sides by :

$\frac{(4 c)}{4} = \frac{- 10}{4}$

Simplify the fraction:

$c = \frac{- 10}{4}$

Find the greatest common factor of the numerator and denominator:

$c = \frac{(- 5 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$c = \frac{- 5}{2}$

3. List the solutions

$c = - 1, - \frac{5}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 c + 6 |$
$y = | c + 4 |$
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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