Solve the Absolute Value Inequality for c |2c+8|=|10c|

$| 2 c + 8 | = | 10 c |$

Step 1

Rewrite the equation as $| 10 c | = | 2 c + 8 |$ .

$| 10 c | = | 2 c + 8 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$10 c = 2 c + 8$

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

$- 1 \cdot (10 c) = 2 c + 8$

$- 1 \cdot (10 c) = - (2 c + 8)$

Step 3

After simplifying, there are only two unique equations to be solved.

$10 c = 2 c + 8$

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

Step 4

Solve $10 c = 2 c + 8$ for $c$ .

Move all terms containing $c$ to the left side of the equation.

Subtract $2 c$ from both sides of the equation.

$10 c - 2 c = 8$

Subtract $2 c$ from $10 c$ .

$8 c = 8$

Divide each term in $8 c = 8$ by $8$ and simplify.

Divide each term in $8 c = 8$ by $8$ .

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

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

Divide $c$ by $1$ .

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

Simplify the right side.

Divide $8$ by $8$ .

$c = 1$

Step 5

Solve $10 c = - (2 c + 8)$ for $c$ .

Simplify $- (2 c + 8)$ .

Apply the distributive property.

$10 c = - (2 c) - 1 \cdot 8$

Multiply.

Multiply $2$ by $- 1$ .

$10 c = - 2 c - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$10 c = - 2 c - 8$

Move all terms containing $c$ to the left side of the equation.

Add $2 c$ to both sides of the equation.

$10 c + 2 c = - 8$

Add $10 c$ and $2 c$ .

$12 c = - 8$

Divide each term in $12 c = - 8$ by $12$ and simplify.

Divide each term in $12 c = - 8$ by $12$ .

$\frac{12 c}{12} = \frac{- 8}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 c}{12} = \frac{- 8}{12}$

Divide $c$ by $1$ .

$c = \frac{- 8}{12}$

Simplify the right side.

Cancel the common factor of $- 8$ and $12$ .

Factor $4$ out of $- 8$ .

$c = \frac{4 (- 2)}{12}$

Cancel the common factors.

Factor $4$ out of $12$ .

$c = \frac{4 \cdot - 2}{4 \cdot 3}$

Cancel the common factor.

$c = \frac{4 \cdot - 2}{4 \cdot 3}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 6

List all of the solutions.

$c = 1, - \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$c = 1, - \frac{2}{3}$

Decimal Form:

$c = 1, - 0.\overline{6}$

Solve the Absolute Value Inequality for c |2c+8|=|10c|

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