Solve the Inequality for x (x+2)^2<=25

${(x + 2)}^{2} \leq 25$

Step 1

Take the square root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{{(x + 2)}^{2}} \leq \sqrt{25}$

Step 2

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x + 2 | \leq \sqrt{25}$

Simplify the right side.

Simplify $\sqrt{25}$ .

Rewrite $25$ as $5^{2}$ .

$| x + 2 | \leq \sqrt{5^{2}}$

Pull terms out from under the radical.

$| x + 2 | \leq | 5 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$| x + 2 | \leq 5$

Step 3

Write $| x + 2 | \leq 5$ as a piecewise.

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

$x + 2 \geq 0$

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

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

$x + 2 \leq 5$

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

$x + 2 < 0$

Subtract $2$ from both sides of the inequality.

$x < - 2$

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

$- (x + 2) \leq 5$

Write as a piecewise.

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

Simplify $- (x + 2) \leq 5$ .

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Step 4

Solve $x + 2 \leq 5$ when $x \geq - 2$ .

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

Subtract $2$ from both sides of the inequality.

$x \leq 5 - 2$

Subtract $2$ from $5$ .

$x \leq 3$

Find the intersection of $x \leq 3$ and $x \geq - 2$ .

$- 2 \leq x \leq 3$

Step 5

Solve $- x - 2 \leq 5$ when $x < - 2$ .

Solve $- x - 2 \leq 5$ for $x$ .

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

Add $2$ to both sides of the inequality.

$- x \leq 5 + 2$

Add $5$ and $2$ .

$- x \leq 7$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $7$ by $- 1$ .

$x \geq - 7$

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

$- 7 \leq x < - 2$

Step 6

Find the union of the solutions.

$- 7 \leq x \leq 3$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$- 7 \leq x \leq 3$

Interval Notation:

$[- 7, 3]$

Step 8

Solve the Inequality for x (x+2)^2<=25

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