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Solution - Absolute value equations

Exact form:

c = 4, - \frac{3}{2}

c=4 , -\frac{3}{2}

Mixed number form:

c = 4, - 1 \frac{1}{2}

c=4 , -1\frac{1}{2}

Decimal form:

c = 4, - 1.5

c=4 , -1.5

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
$| c + 7 | = | 3 c - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| c + 7 \| = \| 3 c - 1 \|$
$x = + y$	$(c + 7) = (3 c - 1)$
$x = - y$	$(c + 7) = - (3 c - 1)$
$+ x = y$	$(c + 7) = (3 c - 1)$
$- x = y$	$- (c + 7) = (3 c - 1)$

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:

$\| x \| = \| y \|$	$\| c + 7 \| = \| 3 c - 1 \|$
$x = + y$ , $+ x = y$	$(c + 7) = (3 c - 1)$
$x = - y$ , $- x = y$	$(c + 7) = - (3 c - 1)$

2. Solve the two equations for c

13 additional steps

$(c + 7) = (3 c - 1)$

Subtract from both sides:

$(c + 7) - 3 c = (3 c - 1) - 3 c$

Group like terms:

$(c - 3 c) + 7 = (3 c - 1) - 3 c$

Simplify the arithmetic:

$- 2 c + 7 = (3 c - 1) - 3 c$

Group like terms:

$- 2 c + 7 = (3 c - 3 c) - 1$

Simplify the arithmetic:

$- 2 c + 7 = - 1$

Subtract from both sides:

$(- 2 c + 7) - 7 = - 1 - 7$

Simplify the arithmetic:

$- 2 c = - 1 - 7$

Simplify the arithmetic:

$- 2 c = - 8$

Divide both sides by :

$\frac{(- 2 c)}{- 2} = \frac{- 8}{- 2}$

Cancel out the negatives:

$\frac{2 c}{2} = \frac{- 8}{- 2}$

Simplify the fraction:

$c = \frac{- 8}{- 2}$

Cancel out the negatives:

$c = \frac{8}{2}$

Find the greatest common factor of the numerator and denominator:

$c = \frac{(4 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$c = 4$

12 additional steps

$(c + 7) = - (3 c - 1)$

Expand the parentheses:

$(c + 7) = - 3 c + 1$

Add to both sides:

$(c + 7) + 3 c = (- 3 c + 1) + 3 c$

Group like terms:

$(c + 3 c) + 7 = (- 3 c + 1) + 3 c$

Simplify the arithmetic:

$4 c + 7 = (- 3 c + 1) + 3 c$

Group like terms:

$4 c + 7 = (- 3 c + 3 c) + 1$

Simplify the arithmetic:

$4 c + 7 = 1$

Subtract from both sides:

$(4 c + 7) - 7 = 1 - 7$

Simplify the arithmetic:

$4 c = 1 - 7$

Simplify the arithmetic:

$4 c = - 6$

Divide both sides by :

$\frac{(4 c)}{4} = \frac{- 6}{4}$

Simplify the fraction:

$c = \frac{- 6}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$c = \frac{- 3}{2}$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | c + 7 |$
$y = | 3 c - 1 |$
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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