Solve for x |4x+9|>11

$| 4 x + 9 | > 11$

Step 1

Write $| 4 x + 9 | > 11$ as a piecewise.

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

$4 x + 9 \geq 0$

Solve the inequality.

Subtract $9$ from both sides of the inequality.

$4 x \geq - 9$

Divide each term in $4 x \geq - 9$ by $4$ and simplify.

Divide each term in $4 x \geq - 9$ by $4$ .

$\frac{4 x}{4} \geq \frac{- 9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{- 9}{4}$

Divide $x$ by $1$ .

$x \geq \frac{- 9}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{9}{4}$

In the piece where $4 x + 9$ is non-negative, remove the absolute value.

$4 x + 9 > 11$

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

$4 x + 9 < 0$

Solve the inequality.

Subtract $9$ from both sides of the inequality.

$4 x < - 9$

Divide each term in $4 x < - 9$ by $4$ and simplify.

Divide each term in $4 x < - 9$ by $4$ .

$\frac{4 x}{4} < \frac{- 9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} < \frac{- 9}{4}$

Divide $x$ by $1$ .

$x < \frac{- 9}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x < - \frac{9}{4}$

In the piece where $4 x + 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x + 9) > 11$

Write as a piecewise.

${\begin{cases} 4 x + 9 > 11 & x \geq - \frac{9}{4} \\ - (4 x + 9) > 11 & x < - \frac{9}{4} \end{cases}$

Simplify $- (4 x + 9) > 11$ .

Apply the distributive property.

${\begin{cases} 4 x + 9 > 11 & x \geq - \frac{9}{4} \\ - (4 x) - 1 \cdot 9 > 11 & x < - \frac{9}{4} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x + 9 > 11 & x \geq - \frac{9}{4} \\ - 4 x - 1 \cdot 9 > 11 & x < - \frac{9}{4} \end{cases}$

Multiply $- 1$ by $9$ .

${\begin{cases} 4 x + 9 > 11 & x \geq - \frac{9}{4} \\ - 4 x - 9 > 11 & x < - \frac{9}{4} \end{cases}$

Step 2

Solve $4 x + 9 > 11$ for $x$ .

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

Subtract $9$ from both sides of the inequality.

$4 x > 11 - 9$

Subtract $9$ from $11$ .

$4 x > 2$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Solve $- 4 x - 9 > 11$ for $x$ .

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

Add $9$ to both sides of the inequality.

$- 4 x > 11 + 9$

Add $11$ and $9$ .

$- 4 x > 20$

Divide each term in $- 4 x > 20$ by $- 4$ and simplify.

Divide each term in $- 4 x > 20$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} < \frac{20}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} < \frac{20}{- 4}$

Divide $x$ by $1$ .

$x < \frac{20}{- 4}$

Simplify the right side.

Divide $20$ by $- 4$ .

$x < - 5$

Step 4

Find the union of the solutions.

$x < - 5$ or $x > \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 5 or x > \frac{1}{2}$

Interval Notation:

$(- \infty, - 5) \cup (\frac{1}{2}, \infty)$

Step 6

Solve for x |4x+9|>11

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