Algebra Examples

$2 \sqrt{x - 8} < 24$

Step 1

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

${(2 \sqrt{x - 8})}^{2} < 24^{2}$

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 $\sqrt{x - 8}$ as ${(x - 8)}^{\frac{1}{2}}$ .

${(2 {(x - 8)}^{\frac{1}{2}})}^{2} < 24^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(2 {(x - 8)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $2 {(x - 8)}^{\frac{1}{2}}$ .

$2^{2} {({(x - 8)}^{\frac{1}{2}})}^{2} < 24^{2}$

Step 2.2.1.2

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

$4 {({(x - 8)}^{\frac{1}{2}})}^{2} < 24^{2}$

Step 2.2.1.3

Multiply the exponents in ${({(x - 8)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.3.1

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

$4 {(x - 8)}^{\frac{1}{2} \cdot 2} < 24^{2}$

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$4 {(x - 8)}^{\frac{1}{2} \cdot 2} < 24^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$4 {(x - 8)}^{1} < 24^{2}$

Step 2.2.1.4

Simplify.

$4 (x - 8) < 24^{2}$

Step 2.2.1.5

Apply the distributive property.

$4 x + 4 \cdot - 8 < 24^{2}$

Step 2.2.1.6

Multiply $4$ by $- 8$ .

$4 x - 32 < 24^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $24$ to the power of $2$ .

$4 x - 32 < 576$

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

Add $32$ to both sides of the inequality.

$4 x < 576 + 32$

Step 3.1.2

Add $576$ and $32$ .

$4 x < 608$

Step 3.2

Divide each term in $4 x < 608$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x < 608$ by $4$ .

$\frac{4 x}{4} < \frac{608}{4}$

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.

$\frac{4 x}{4} < \frac{608}{4}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{608}{4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $608$ by $4$ .

$x < 152$

Step 4

Find the domain of $2 \sqrt{x - 8} - 24$ .

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Step 4.1

Set the radicand in $\sqrt{x - 8}$ greater than or equal to $0$ to find where the expression is defined.

$x - 8 \geq 0$

Step 4.2

Add $8$ to both sides of the inequality.

$x \geq 8$

Step 4.3

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

$[8, \infty)$

Step 5

Use each root to create test intervals.

$x < 8$

$8 < x < 152$

$x > 152$

Step 6

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 6.1

Test a value on the interval $x < 8$ to see if it makes the inequality true.

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Step 6.1.1

Choose a value on the interval $x < 8$ and see if this value makes the original inequality true.

$x = 0$

Step 6.1.2

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

$2 \sqrt{(0) - 8} < 24$

Step 6.1.3

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

False

Step 6.2

Test a value on the interval $8 < x < 152$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $8 < x < 152$ and see if this value makes the original inequality true.

$x = 10$

Step 6.2.2

Replace $x$ with $10$ in the original inequality.

$2 \sqrt{(10) - 8} < 24$

Step 6.2.3

The left side $2.82842712$ is less than the right side $24$ , which means that the given statement is always true.

True

Step 6.3

Test a value on the interval $x > 152$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $x > 152$ and see if this value makes the original inequality true.

$x = 154$

Step 6.3.2

Replace $x$ with $154$ in the original inequality.

$2 \sqrt{(154) - 8} < 24$

Step 6.3.3

The left side $24.16609194$ is not less than the right side $24$ , which means that the given statement is false.

False

Step 6.4

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

$x < 8$ False

$8 < x < 152$ True

$x > 152$ False

$x < 8$ False

$8 < x < 152$ True

$x > 152$ False

Step 7

The solution consists of all of the true intervals.

$8 < x < 152$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$8 < x < 152$

Interval Notation:

$(8, 152)$

Step 9