Precalculus Examples

$\sqrt{{(2 x + 5)}^{2}} < 11$

Step 1

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{{(2 x + 5)}^{2}}}^{2} < 11^{2}$

Step 2

Simplify each side of the inequality.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(2 x + 5)}^{2}}$ as ${(2 x + 5)}^{\frac{2}{2}}$ .

${({(2 x + 5)}^{\frac{2}{2}})}^{2} < 11^{2}$

Step 2.2

Divide $2$ by $2$ .

${({(2 x + 5)}^{1})}^{2} < 11^{2}$

Step 2.3

Simplify the left side.

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Step 2.3.1

Multiply the exponents in ${({(2 x + 5)}^{1})}^{2}$ .

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Step 2.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x + 5)}^{1 \cdot 2} < 11^{2}$

Step 2.3.1.2

Multiply $2$ by $1$ .

${(2 x + 5)}^{2} < 11^{2}$

Step 2.4

Simplify the right side.

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Step 2.4.1

Raise $11$ to the power of $2$ .

${(2 x + 5)}^{2} < 121$

Step 3

Solve for $x$ .

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Step 3.1

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

$\sqrt{{(2 x + 5)}^{2}} < \sqrt{121}$

Step 3.2

Simplify the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Pull terms out from under the radical.

$| 2 x + 5 | < \sqrt{121}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\sqrt{121}$ .

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Step 3.2.2.1.1

Rewrite $121$ as $11^{2}$ .

$| 2 x + 5 | < \sqrt{11^{2}}$

Step 3.2.2.1.2

Pull terms out from under the radical.

$| 2 x + 5 | < | 11 |$

Step 3.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $11$ is $11$ .

$| 2 x + 5 | < 11$

Step 3.3

Write $| 2 x + 5 | < 11$ as a piecewise.

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Step 3.3.1

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

$2 x + 5 \geq 0$

Step 3.3.2

Solve the inequality.

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Step 3.3.2.1

Subtract $5$ from both sides of the inequality.

$2 x \geq - 5$

Step 3.3.2.2

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

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Step 3.3.2.2.1

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

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

Step 3.3.2.2.2

Simplify the left side.

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Step 3.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.2.2.1.1

Cancel the common factor.

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

Step 3.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.2.2.3

Simplify the right side.

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Step 3.3.2.2.3.1

Move the negative in front of the fraction.

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

Step 3.3.3

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

$2 x + 5 < 11$

Step 3.3.4

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

$2 x + 5 < 0$

Step 3.3.5

Solve the inequality.

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Step 3.3.5.1

Subtract $5$ from both sides of the inequality.

$2 x < - 5$

Step 3.3.5.2

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

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Step 3.3.5.2.1

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

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

Step 3.3.5.2.2

Simplify the left side.

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Step 3.3.5.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.5.2.2.1.1

Cancel the common factor.

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

Step 3.3.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.5.2.3

Simplify the right side.

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Step 3.3.5.2.3.1

Move the negative in front of the fraction.

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

Step 3.3.6

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

$- (2 x + 5) < 11$

Step 3.3.7

Write as a piecewise.

${\begin{cases} 2 x + 5 < 11 & x \geq - \frac{5}{2} \\ - (2 x + 5) < 11 & x < - \frac{5}{2} \end{cases}$

Step 3.3.8

Simplify $- (2 x + 5) < 11$ .

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Step 3.3.8.1

Apply the distributive property.

${\begin{cases} 2 x + 5 < 11 & x \geq - \frac{5}{2} \\ - (2 x) - 1 \cdot 5 < 11 & x < - \frac{5}{2} \end{cases}$

Step 3.3.8.2

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x + 5 < 11 & x \geq - \frac{5}{2} \\ - 2 x - 1 \cdot 5 < 11 & x < - \frac{5}{2} \end{cases}$

Step 3.3.8.3

Multiply $- 1$ by $5$ .

${\begin{cases} 2 x + 5 < 11 & x \geq - \frac{5}{2} \\ - 2 x - 5 < 11 & x < - \frac{5}{2} \end{cases}$

Step 3.4

Solve $2 x + 5 < 11$ when $x \geq - \frac{5}{2}$ .

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Step 3.4.1

Solve $2 x + 5 < 11$ for $x$ .

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Step 3.4.1.1

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

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Step 3.4.1.1.1

Subtract $5$ from both sides of the inequality.

$2 x < 11 - 5$

Step 3.4.1.1.2

Subtract $5$ from $11$ .

$2 x < 6$

Step 3.4.1.2

Divide each term in $2 x < 6$ by $2$ and simplify.

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Step 3.4.1.2.1

Divide each term in $2 x < 6$ by $2$ .

$\frac{2 x}{2} < \frac{6}{2}$

Step 3.4.1.2.2

Simplify the left side.

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Step 3.4.1.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{6}{2}$

Step 3.4.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{6}{2}$

Step 3.4.1.2.3

Simplify the right side.

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Step 3.4.1.2.3.1

Divide $6$ by $2$ .

$x < 3$

Step 3.4.2

Find the intersection of $x < 3$ and $x \geq - \frac{5}{2}$ .

$- \frac{5}{2} \leq x < 3$

Step 3.5

Solve $- 2 x - 5 < 11$ when $x < - \frac{5}{2}$ .

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Step 3.5.1

Solve $- 2 x - 5 < 11$ for $x$ .

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Step 3.5.1.1

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

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Step 3.5.1.1.1

Add $5$ to both sides of the inequality.

$- 2 x < 11 + 5$

Step 3.5.1.1.2

Add $11$ and $5$ .

$- 2 x < 16$

Step 3.5.1.2

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

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Step 3.5.1.2.1

Divide each term in $- 2 x < 16$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{16}{- 2}$

Step 3.5.1.2.2

Simplify the left side.

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Step 3.5.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.5.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{16}{- 2}$

Step 3.5.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{16}{- 2}$

Step 3.5.1.2.3

Simplify the right side.

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Step 3.5.1.2.3.1

Divide $16$ by $- 2$ .

$x > - 8$

Step 3.5.2

Find the intersection of $x > - 8$ and $x < - \frac{5}{2}$ .

$- 8 < x < - \frac{5}{2}$

Step 3.6

Find the union of the solutions.

$- 8 < x < 3$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 8 < x < 3$

Interval Notation:

$(- 8, 3)$

Step 5