Solve for x |2x+9|<25

$| 2 x + 9 | < 25$

Step 1

Write $| 2 x + 9 | < 25$ as a piecewise.

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

$2 x + 9 \geq 0$

Solve the inequality.

Subtract $9$ from both sides of the inequality.

$2 x \geq - 9$

Divide each term in $2 x \geq - 9$ by $2$ and simplify.

Divide each term in $2 x \geq - 9$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 9}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 9}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 9}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{9}{2}$

In the piece where $2 x + 9$ is non-negative, remove the absolute value.

$2 x + 9 < 25$

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

$2 x + 9 < 0$

Solve the inequality.

Subtract $9$ from both sides of the inequality.

$2 x < - 9$

Divide each term in $2 x < - 9$ by $2$ and simplify.

Divide each term in $2 x < - 9$ by $2$ .

$\frac{2 x}{2} < \frac{- 9}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 9}{2}$

Divide $x$ by $1$ .

$x < \frac{- 9}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x < - \frac{9}{2}$

In the piece where $2 x + 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x + 9) < 25$

Write as a piecewise.

${\begin{cases} 2 x + 9 < 25 & x \geq - \frac{9}{2} \\ - (2 x + 9) < 25 & x < - \frac{9}{2} \end{cases}$

Simplify $- (2 x + 9) < 25$ .

Apply the distributive property.

${\begin{cases} 2 x + 9 < 25 & x \geq - \frac{9}{2} \\ - (2 x) - 1 \cdot 9 < 25 & x < - \frac{9}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x + 9 < 25 & x \geq - \frac{9}{2} \\ - 2 x - 1 \cdot 9 < 25 & x < - \frac{9}{2} \end{cases}$

Multiply $- 1$ by $9$ .

${\begin{cases} 2 x + 9 < 25 & x \geq - \frac{9}{2} \\ - 2 x - 9 < 25 & x < - \frac{9}{2} \end{cases}$

Step 2

Solve $2 x + 9 < 25$ when $x \geq - \frac{9}{2}$ .

Solve $2 x + 9 < 25$ for $x$ .

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

Subtract $9$ from both sides of the inequality.

$2 x < 25 - 9$

Subtract $9$ from $25$ .

$2 x < 16$

Divide each term in $2 x < 16$ by $2$ and simplify.

Divide each term in $2 x < 16$ by $2$ .

$\frac{2 x}{2} < \frac{16}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{16}{2}$

Divide $x$ by $1$ .

$x < \frac{16}{2}$

Simplify the right side.

Divide $16$ by $2$ .

$x < 8$

Find the intersection of $x < 8$ and $x \geq - \frac{9}{2}$ .

$- \frac{9}{2} \leq x < 8$

Step 3

Solve $- 2 x - 9 < 25$ when $x < - \frac{9}{2}$ .

Solve $- 2 x - 9 < 25$ for $x$ .

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

Add $9$ to both sides of the inequality.

$- 2 x < 25 + 9$

Add $25$ and $9$ .

$- 2 x < 34$

Divide each term in $- 2 x < 34$ by $- 2$ and simplify.

Divide each term in $- 2 x < 34$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{34}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{34}{- 2}$

Divide $x$ by $1$ .

$x > \frac{34}{- 2}$

Simplify the right side.

Divide $34$ by $- 2$ .

$x > - 17$

Find the intersection of $x > - 17$ and $x < - \frac{9}{2}$ .

$- 17 < x < - \frac{9}{2}$

Step 4

Find the union of the solutions.

$- 17 < x < 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 17 < x < 8$

Interval Notation:

$(- 17, 8)$

Step 6

Solve for x |2x+9|<25

Related Questions

Support LCMGCF.com by disabling your adblocker.