Algebra Examples

$9 - \sqrt{c + 4} \leq 6$

Step 1

Move all terms not containing $- \sqrt{c + 4}$ to the right side of the inequality.

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Step 1.1

Subtract $9$ from both sides of the inequality.

$- \sqrt{c + 4} \leq 6 - 9$

Step 1.2

Subtract $9$ from $6$ .

$- \sqrt{c + 4} \leq - 3$

Step 2

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

${(- \sqrt{c + 4})}^{2} \leq {(- 3)}^{2}$

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 $\sqrt{c + 4}$ as ${(c + 4)}^{\frac{1}{2}}$ .

${(- {(c + 4)}^{\frac{1}{2}})}^{2} \leq {(- 3)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- {(c + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- {(c + 4)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(c + 4)}^{\frac{1}{2}})}^{2} \leq {(- 3)}^{2}$

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(c + 4)}^{\frac{1}{2}})}^{2} \leq {(- 3)}^{2}$

Step 3.2.1.3

Multiply ${({(c + 4)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(c + 4)}^{\frac{1}{2}})}^{2} \leq {(- 3)}^{2}$

Step 3.2.1.4

Multiply the exponents in ${({(c + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.4.1

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

${(c + 4)}^{\frac{1}{2} \cdot 2} \leq {(- 3)}^{2}$

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

${(c + 4)}^{\frac{1}{2} \cdot 2} \leq {(- 3)}^{2}$

Step 3.2.1.4.2.2

Rewrite the expression.

${(c + 4)}^{1} \leq {(- 3)}^{2}$

Step 3.2.1.5

Simplify.

$c + 4 \leq {(- 3)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $- 3$ to the power of $2$ .

$c + 4 \leq 9$

Step 4

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

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Step 4.1

Subtract $4$ from both sides of the inequality.

$c \leq 9 - 4$

Step 4.2

Subtract $4$ from $9$ .

$c \leq 5$

Step 5

Find the domain of $3 - \sqrt{c + 4}$ .

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Step 5.1

Set the radicand in $\sqrt{c + 4}$ greater than or equal to $0$ to find where the expression is defined.

$c + 4 \geq 0$

Step 5.2

Subtract $4$ from both sides of the inequality.

$c \geq - 4$

Step 5.3

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

$[- 4, \infty)$

Step 6

The solution consists of all of the true intervals.

$c \geq 5$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$c \geq 5$

Interval Notation:

$[5, \infty)$

Step 8