Solve the Inequality for z square root of z+5+4<=13

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

Step 1

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

Subtract $4$ from both sides of the inequality.

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

Subtract $4$ from $13$ .

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

Step 2

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

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

Step 3

Simplify each side of the inequality.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$z + 5 \leq 81$

Step 4

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

Subtract $5$ from both sides of the inequality.

$z \leq 81 - 5$

Subtract $5$ from $81$ .

$z \leq 76$

Step 5

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

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

$z + 5 \geq 0$

Subtract $5$ from both sides of the inequality.

$z \geq - 5$

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

$[- 5, \infty)$

Step 6

Use each root to create test intervals.

$z < - 5$

$- 5 < z < 76$

$z > 76$

Step 7

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

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

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

$z = - 8$

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

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

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

False

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

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

$z = 0$

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

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

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

True

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

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

$z = 78$

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

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

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

False

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

$z < - 5$ False

$- 5 < z < 76$ True

$z > 76$ False

$z < - 5$ False

$- 5 < z < 76$ True

$z > 76$ False

Step 8

The solution consists of all of the true intervals.

$- 5 \leq z \leq 76$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq z \leq 76$

Interval Notation:

$[- 5, 76]$

Step 10

Solve the Inequality for z square root of z+5+4<=13

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