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Solution - Absolute value equations

Exact form:

c = 10, - 2

c=10 , -2

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
$| 5 c - 2 | = 3 | c + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 c - 2 \| = 3 \| c + 6 \|$
$x = + y$	$(5 c - 2) = 3 (c + 6)$
$x = - y$	$(5 c - 2) = 3 (- (c + 6))$
$+ x = y$	$(5 c - 2) = 3 (c + 6)$
$- x = y$	$- (5 c - 2) = 3 (c + 6)$

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:

$\| x \| = \| y \|$	$\| 5 c - 2 \| = 3 \| c + 6 \|$
$x = + y$ , $+ x = y$	$(5 c - 2) = 3 (c + 6)$
$x = - y$ , $- x = y$	$(5 c - 2) = 3 (- (c + 6))$

2. Solve the two equations for c

13 additional steps

$(5 c - 2) = 3 \cdot (c + 6)$

Expand the parentheses:

$(5 c - 2) = 3 c + 3 \cdot 6$

Simplify the arithmetic:

$(5 c - 2) = 3 c + 18$

Subtract from both sides:

$(5 c - 2) - 3 c = (3 c + 18) - 3 c$

Group like terms:

$(5 c - 3 c) - 2 = (3 c + 18) - 3 c$

Simplify the arithmetic:

$2 c - 2 = (3 c + 18) - 3 c$

Group like terms:

$2 c - 2 = (3 c - 3 c) + 18$

Simplify the arithmetic:

$2 c - 2 = 18$

Add to both sides:

$(2 c - 2) + 2 = 18 + 2$

Simplify the arithmetic:

$2 c = 18 + 2$

Simplify the arithmetic:

$2 c = 20$

Divide both sides by :

$\frac{(2 c)}{2} = \frac{20}{2}$

Simplify the fraction:

$c = \frac{20}{2}$

Find the greatest common factor of the numerator and denominator:

$c = \frac{(10 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$c = 10$

16 additional steps

$(5 c - 2) = 3 \cdot (- (c + 6))$

Expand the parentheses:

$(5 c - 2) = 3 \cdot (- c - 6)$

$(5 c - 2) = 3 \cdot - c + 3 \cdot - 6$

Group like terms:

$(5 c - 2) = (3 \cdot - 1) c + 3 \cdot - 6$

Multiply the coefficients:

$(5 c - 2) = - 3 c + 3 \cdot - 6$

Simplify the arithmetic:

$(5 c - 2) = - 3 c - 18$

Add to both sides:

$(5 c - 2) + 3 c = (- 3 c - 18) + 3 c$

Group like terms:

$(5 c + 3 c) - 2 = (- 3 c - 18) + 3 c$

Simplify the arithmetic:

$8 c - 2 = (- 3 c - 18) + 3 c$

Group like terms:

$8 c - 2 = (- 3 c + 3 c) - 18$

Simplify the arithmetic:

$8 c - 2 = - 18$

Add to both sides:

$(8 c - 2) + 2 = - 18 + 2$

Simplify the arithmetic:

$8 c = - 18 + 2$

Simplify the arithmetic:

$8 c = - 16$

Divide both sides by :

$\frac{(8 c)}{8} = \frac{- 16}{8}$

Simplify the fraction:

$c = \frac{- 16}{8}$

Find the greatest common factor of the numerator and denominator:

$c = \frac{(- 2 \cdot 8)}{(1 \cdot 8)}$

Factor out and cancel the greatest common factor:

$c = - 2$

3. List the solutions

$c = 10, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 c - 2 |$
$y = 3 | c + 6 |$
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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