Solve the Inequality for x |5-2x|>9

$| 5 - 2 x | > 9$

Step 1

Write $| 5 - 2 x | > 9$ as a piecewise.

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

$5 - 2 x \geq 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$- 2 x \geq - 5$

Divide each term in $- 2 x \geq - 5$ by $- 2$ and simplify.

Divide each term in $- 2 x \geq - 5$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 5}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 5}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{5}{2}$

In the piece where $5 - 2 x$ is non-negative, remove the absolute value.

$5 - 2 x > 9$

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

$5 - 2 x < 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$- 2 x < - 5$

Divide each term in $- 2 x < - 5$ by $- 2$ and simplify.

Divide each term in $- 2 x < - 5$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{- 5}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{- 5}{- 2}$

Divide $x$ by $1$ .

$x > \frac{- 5}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{5}{2}$

In the piece where $5 - 2 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 - 2 x) > 9$

Write as a piecewise.

${\begin{cases} 5 - 2 x > 9 & x \leq \frac{5}{2} \\ - (5 - 2 x) > 9 & x > \frac{5}{2} \end{cases}$

Simplify $- (5 - 2 x) > 9$ .

Apply the distributive property.

${\begin{cases} 5 - 2 x > 9 & x \leq \frac{5}{2} \\ - 1 \cdot 5 - (- 2 x) > 9 & x > \frac{5}{2} \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 - 2 x > 9 & x \leq \frac{5}{2} \\ - 5 - (- 2 x) > 9 & x > \frac{5}{2} \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} 5 - 2 x > 9 & x \leq \frac{5}{2} \\ - 5 + 2 x > 9 & x > \frac{5}{2} \end{cases}$

Step 2

Solve $5 - 2 x > 9$ for $x$ .

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

Subtract $5$ from both sides of the inequality.

$- 2 x > 9 - 5$

Subtract $5$ from $9$ .

$- 2 x > 4$

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

Divide each term in $- 2 x > 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{4}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{4}{- 2}$

Divide $x$ by $1$ .

$x < \frac{4}{- 2}$

Simplify the right side.

Divide $4$ by $- 2$ .

$x < - 2$

Step 3

Solve $- 5 + 2 x > 9$ for $x$ .

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

Add $5$ to both sides of the inequality.

$2 x > 9 + 5$

Add $9$ and $5$ .

$2 x > 14$

Divide each term in $2 x > 14$ by $2$ and simplify.

Divide each term in $2 x > 14$ by $2$ .

$\frac{2 x}{2} > \frac{14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{14}{2}$

Divide $x$ by $1$ .

$x > \frac{14}{2}$

Simplify the right side.

Divide $14$ by $2$ .

$x > 7$

Step 4

Find the union of the solutions.

$x < - 2$ or $x > 7$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 7$

Interval Notation:

$(- \infty, - 2) \cup (7, \infty)$

Step 6

Solve the Inequality for x |5-2x|>9

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