Algebra Examples

$\sqrt{x + 5} + 4 \leq 13$

Step 1

Move all terms not containing $\sqrt{x + 5}$ to the right side of the inequality.

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Step 1.1

Subtract $4$ from both sides of the inequality.

$\sqrt{x + 5} \leq 13 - 4$

Step 1.2

Subtract $4$ from $13$ .

$\sqrt{x + 5} \leq 9$

Step 2

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

${\sqrt{x + 5}}^{2} \leq 9^{2}$

Step 3

Simplify each side of the inequality.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 5}$ as ${(x + 5)}^{\frac{1}{2}}$ .

${({(x + 5)}^{\frac{1}{2}})}^{2} \leq 9^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(x + 5)}^{\frac{1}{2} \cdot 2} \leq 9^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(x + 5)}^{\frac{1}{2} \cdot 2} \leq 9^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 5)}^{1} \leq 9^{2}$

Step 3.2.1.2

Simplify.

$x + 5 \leq 9^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $9$ to the power of $2$ .

$x + 5 \leq 81$

Step 4

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

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Step 4.1

Subtract $5$ from both sides of the inequality.

$x \leq 81 - 5$

Step 4.2

Subtract $5$ from $81$ .

$x \leq 76$

Step 5

Find the domain of $\sqrt{x + 5} - 9$ .

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Step 5.1

Set the radicand in $\sqrt{x + 5}$ greater than or equal to $0$ to find where the expression is defined.

$x + 5 \geq 0$

Step 5.2

Subtract $5$ from both sides of the inequality.

$x \geq - 5$

Step 5.3

The domain is all values of $x$ that make the expression defined.

$[- 5, \infty)$

Step 6

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 76$

$x > 76$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $x < - 5$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

Step 7.1.2

Replace $x$ with $- 8$ in the original inequality.

$\sqrt{(- 8) + 5} + 4 \leq 13$

Step 7.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- 5 < x < 76$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 5 < x < 76$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

Replace $x$ with $0$ in the original inequality.

$\sqrt{(0) + 5} + 4 \leq 13$

Step 7.2.3

The left side $6.23606797$ is less than the right side $13$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $x > 76$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > 76$ and see if this value makes the original inequality true.

$x = 78$

Step 7.3.2

Replace $x$ with $78$ in the original inequality.

$\sqrt{(78) + 5} + 4 \leq 13$

Step 7.3.3

The left side $13.11043357$ is greater than the right side $13$ , which means that the given statement is false.

False

Step 7.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 5$ False

$- 5 < x < 76$ True

$x > 76$ False

$x < - 5$ False

$- 5 < x < 76$ True

$x > 76$ False

Step 8

The solution consists of all of the true intervals.

$- 5 \leq x \leq 76$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq x \leq 76$

Interval Notation:

$[- 5, 76]$

Step 10