Solve for c |c+4|<5

$| c + 4 | < 5$

Step 1

Write $| c + 4 | < 5$ as a piecewise.

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

$c + 4 \geq 0$

Subtract $4$ from both sides of the inequality.

$c \geq - 4$

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

$c + 4 < 5$

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

$c + 4 < 0$

Subtract $4$ from both sides of the inequality.

$c < - 4$

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

$- (c + 4) < 5$

Write as a piecewise.

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

Simplify $- (c + 4) < 5$ .

Apply the distributive property.

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

Multiply $- 1$ by $4$ .

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

Step 2

Solve $c + 4 < 5$ when $c \geq - 4$ .

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

Subtract $4$ from both sides of the inequality.

$c < 5 - 4$

Subtract $4$ from $5$ .

$c < 1$

Find the intersection of $c < 1$ and $c \geq - 4$ .

$- 4 \leq c < 1$

Step 3

Solve $- c - 4 < 5$ when $c < - 4$ .

Solve $- c - 4 < 5$ for $c$ .

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

Add $4$ to both sides of the inequality.

$- c < 5 + 4$

Add $5$ and $4$ .

$- c < 9$

Divide each term in $- c < 9$ by $- 1$ and simplify.

Divide each term in $- c < 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- c}{- 1} > \frac{9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{c}{1} > \frac{9}{- 1}$

Divide $c$ by $1$ .

$c > \frac{9}{- 1}$

Simplify the right side.

Divide $9$ by $- 1$ .

$c > - 9$

Find the intersection of $c > - 9$ and $c < - 4$ .

$- 9 < c < - 4$

Step 4

Find the union of the solutions.

$- 9 < c < 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 9 < c < 1$

Interval Notation:

$(- 9, 1)$

Step 6

Solve for c |c+4|<5

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