Solve for c |4c+5|>7

$| 4 c + 5 | > 7$

Write $| 4 c + 5 | > 7$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$4 c + 5 \geq 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$4 c \geq - 5$

Divide each term in $4 c \geq - 5$ by $4$ and simplify.

Divide each term in $4 c \geq - 5$ by $4$ .

$\frac{4 c}{4} \geq \frac{- 5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 c}{4} \geq \frac{- 5}{4}$

Divide $c$ by $1$ .

$c \geq \frac{- 5}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$c \geq - \frac{5}{4}$

In the piece where $4 c + 5$ is non-negative, remove the absolute value.

$4 c + 5 > 7$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$4 c + 5 < 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$4 c < - 5$

Divide each term in $4 c < - 5$ by $4$ and simplify.

Divide each term in $4 c < - 5$ by $4$ .

$\frac{4 c}{4} < \frac{- 5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 c}{4} < \frac{- 5}{4}$

Divide $c$ by $1$ .

$c < \frac{- 5}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$c < - \frac{5}{4}$

In the piece where $4 c + 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 c + 5) > 7$

Write as a piecewise.

${\begin{cases} 4 c + 5 > 7 & c \geq - \frac{5}{4} \\ - (4 c + 5) > 7 & c < - \frac{5}{4} \end{cases}$

Simplify $- (4 c + 5) > 7$ .

Apply the distributive property.

${\begin{cases} 4 c + 5 > 7 & c \geq - \frac{5}{4} \\ - (4 c) - 1 \cdot 5 > 7 & c < - \frac{5}{4} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 4 c + 5 > 7 & c \geq - \frac{5}{4} \\ - 4 c - 1 \cdot 5 > 7 & c < - \frac{5}{4} \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 4 c + 5 > 7 & c \geq - \frac{5}{4} \\ - 4 c - 5 > 7 & c < - \frac{5}{4} \end{cases}$

Solve $4 c + 5 > 7$ for $c$ .

Move all terms not containing $c$ to the right side of the inequality.

Subtract $5$ from both sides of the inequality.

$4 c > 7 - 5$

Subtract $5$ from $7$ .

$4 c > 2$

Divide each term in $4 c > 2$ by $4$ and simplify.

Divide each term in $4 c > 2$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $c$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$c > \frac{2 (1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$c > \frac{2 \cdot 1}{2 \cdot 2}$

Cancel the common factor.

$c > \frac{2 \cdot 1}{2 \cdot 2}$

Rewrite the expression.

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

Solve $- 4 c - 5 > 7$ for $c$ .

Move all terms not containing $c$ to the right side of the inequality.

Add $5$ to both sides of the inequality.

$- 4 c > 7 + 5$

Add $7$ and $5$ .

$- 4 c > 12$

Divide each term in $- 4 c > 12$ by $- 4$ and simplify.

Divide each term in $- 4 c > 12$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 c}{- 4} < \frac{12}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 c}{- 4} < \frac{12}{- 4}$

Divide $c$ by $1$ .

$c < \frac{12}{- 4}$

Simplify the right side.

Divide $12$ by $- 4$ .

$c < - 3$

Find the union of the solutions.

$c < - 3$ or $c > \frac{1}{2}$

The result can be shown in multiple forms.

Inequality Form:

$c < - 3 or c > \frac{1}{2}$

Interval Notation:

$(- \infty, - 3) \cup (\frac{1}{2}, \infty)$

Solve for c |4c+5|>7

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