Solve for x |7+2x|<11

$| 7 + 2 x | < 11$

Step 1

Write $| 7 + 2 x | < 11$ as a piecewise.

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

$7 + 2 x \geq 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$2 x \geq - 7$

Divide each term in $2 x \geq - 7$ by $2$ and simplify.

Divide each term in $2 x \geq - 7$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 7}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{7}{2}$

In the piece where $7 + 2 x$ is non-negative, remove the absolute value.

$7 + 2 x < 11$

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

$7 + 2 x < 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$2 x < - 7$

Divide each term in $2 x < - 7$ by $2$ and simplify.

Divide each term in $2 x < - 7$ by $2$ .

$\frac{2 x}{2} < \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 7}{2}$

Divide $x$ by $1$ .

$x < \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x < - \frac{7}{2}$

In the piece where $7 + 2 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (7 + 2 x) < 11$

Write as a piecewise.

${\begin{cases} 7 + 2 x < 11 & x \geq - \frac{7}{2} \\ - (7 + 2 x) < 11 & x < - \frac{7}{2} \end{cases}$

Simplify $- (7 + 2 x) < 11$ .

Apply the distributive property.

${\begin{cases} 7 + 2 x < 11 & x \geq - \frac{7}{2} \\ - 1 \cdot 7 - (2 x) < 11 & x < - \frac{7}{2} \end{cases}$

Multiply $- 1$ by $7$ .

${\begin{cases} 7 + 2 x < 11 & x \geq - \frac{7}{2} \\ - 7 - (2 x) < 11 & x < - \frac{7}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 7 + 2 x < 11 & x \geq - \frac{7}{2} \\ - 7 - 2 x < 11 & x < - \frac{7}{2} \end{cases}$

Step 2

Solve $7 + 2 x < 11$ when $x \geq - \frac{7}{2}$ .

Solve $7 + 2 x < 11$ for $x$ .

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

Subtract $7$ from both sides of the inequality.

$2 x < 11 - 7$

Subtract $7$ from $11$ .

$2 x < 4$

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

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

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

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 $2$ .

$x < 2$

Find the intersection of $x < 2$ and $x \geq - \frac{7}{2}$ .

$- \frac{7}{2} \leq x < 2$

Step 3

Solve $- 7 - 2 x < 11$ when $x < - \frac{7}{2}$ .

Solve $- 7 - 2 x < 11$ for $x$ .

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

Add $7$ to both sides of the inequality.

$- 2 x < 11 + 7$

Add $11$ and $7$ .

$- 2 x < 18$

Divide each term in $- 2 x < 18$ by $- 2$ and simplify.

Divide each term in $- 2 x < 18$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{18}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{18}{- 2}$

Divide $x$ by $1$ .

$x > \frac{18}{- 2}$

Simplify the right side.

Divide $18$ by $- 2$ .

$x > - 9$

Find the intersection of $x > - 9$ and $x < - \frac{7}{2}$ .

$- 9 < x < - \frac{7}{2}$

Step 4

Find the union of the solutions.

$- 9 < x < 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < 2$

Interval Notation:

$(- 9, 2)$

Step 6

Solve for x |7+2x|<11

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