Solve the Inequality for x |2x+5|<2-x

$| 2 x + 5 | < 2 - x$

Write $| 2 x + 5 | < 2 - x$ as a piecewise.

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

$2 x + 5 \geq 0$

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 $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

$2 x + 5 < 2 - x$

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

$2 x + 5 < 0$

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 $2$ .

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

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.

Move the negative in front of the fraction.

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

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

$- (2 x + 5) < 2 - x$

Write as a piecewise.

${\begin{cases} 2 x + 5 < 2 - x & x \geq - \frac{5}{2} \\ - (2 x + 5) < 2 - x & x < - \frac{5}{2} \end{cases}$

Simplify $- (2 x + 5) < 2 - x$ .

Apply the distributive property.

${\begin{cases} 2 x + 5 < 2 - x & x \geq - \frac{5}{2} \\ - (2 x) - 1 \cdot 5 < 2 - x & x < - \frac{5}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x + 5 < 2 - x & x \geq - \frac{5}{2} \\ - 2 x - 1 \cdot 5 < 2 - x & x < - \frac{5}{2} \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 2 x + 5 < 2 - x & x \geq - \frac{5}{2} \\ - 2 x - 5 < 2 - x & x < - \frac{5}{2} \end{cases}$

Solve $2 x + 5 < 2 - x$ when $x \geq - \frac{5}{2}$ .

Solve $2 x + 5 < 2 - x$ for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Add $x$ to both sides of the inequality.

$2 x + 5 + x < 2$

Add $2 x$ and $x$ .

$3 x + 5 < 2$

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

Subtract $5$ from both sides of the inequality.

$3 x < 2 - 5$

Subtract $5$ from $2$ .

$3 x < - 3$

Divide each term in $3 x < - 3$ by $3$ and simplify.

Divide each term in $3 x < - 3$ by $3$ .

$\frac{3 x}{3} < \frac{- 3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 3}{3}$

Divide $x$ by $1$ .

$x < \frac{- 3}{3}$

Simplify the right side.

Divide $- 3$ by $3$ .

$x < - 1$

Find the intersection of $x < - 1$ and $x \geq - \frac{5}{2}$ .

$- \frac{5}{2} \leq x < - 1$

Solve $- 2 x - 5 < 2 - x$ when $x < - \frac{5}{2}$ .

Solve $- 2 x - 5 < 2 - x$ for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Add $x$ to both sides of the inequality.

$- 2 x - 5 + x < 2$

Add $- 2 x$ and $x$ .

$- x - 5 < 2$

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

Add $5$ to both sides of the inequality.

$- x < 2 + 5$

Add $2$ and $5$ .

$- x < 7$

Divide each term in $- x < 7$ by $- 1$ and simplify.

Divide each term in $- x < 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{7}{- 1}$

Divide $x$ by $1$ .

$x > \frac{7}{- 1}$

Simplify the right side.

Divide $7$ by $- 1$ .

$x > - 7$

Find the intersection of $x > - 7$ and $x < - \frac{5}{2}$ .

$- 7 < x < - \frac{5}{2}$

Find the union of the solutions.

$- 7 < x < - 1$

The result can be shown in multiple forms.

Inequality Form:

$- 7 < x < - 1$

Interval Notation:

$(- 7, - 1)$

Solve the Inequality for x |2x+5|<2-x

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