Algebra Examples

$\sqrt{4 x + 28} > 8$

Step 1

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

${\sqrt{4 x + 28}}^{2} > 8^{2}$

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 $\sqrt{4 x + 28}$ as ${(4 x + 28)}^{\frac{1}{2}}$ .

${({(4 x + 28)}^{\frac{1}{2}})}^{2} > 8^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(4 x + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(4 x + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(4 x + 28)}^{\frac{1}{2} \cdot 2} > 8^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(4 x + 28)}^{\frac{1}{2} \cdot 2} > 8^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(4 x + 28)}^{1} > 8^{2}$

Step 2.2.1.2

Simplify.

$4 x + 28 > 8^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $8$ to the power of $2$ .

$4 x + 28 > 64$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $28$ from both sides of the inequality.

$4 x > 64 - 28$

Step 3.1.2

Subtract $28$ from $64$ .

$4 x > 36$

Step 3.2

Divide each term in $4 x > 36$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x > 36$ by $4$ .

$\frac{4 x}{4} > \frac{36}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} > \frac{36}{4}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{36}{4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $36$ by $4$ .

$x > 9$

Step 4

Find the domain of $2 \sqrt{x + 7} - 8$ .

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Step 4.1

Set the radicand in $\sqrt{x + 7}$ greater than or equal to $0$ to find where the expression is defined.

$x + 7 \geq 0$

Step 4.2

Subtract $7$ from both sides of the inequality.

$x \geq - 7$

Step 4.3

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

$[- 7, \infty)$

Step 5

The solution consists of all of the true intervals.

$x > 9$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > 9$

Interval Notation:

$(9, \infty)$

Step 7