Algebra Examples

$- 18 \sqrt{2 x - 1} > - 36$

Step 1

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

${(- 18 \sqrt{2 x - 1})}^{2} > {(- 36)}^{2}$

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

${(- 18 {(2 x - 1)}^{\frac{1}{2}})}^{2} > {(- 36)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(- 18 {(2 x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $- 18 {(2 x - 1)}^{\frac{1}{2}}$ .

${(- 18)}^{2} {({(2 x - 1)}^{\frac{1}{2}})}^{2} > {(- 36)}^{2}$

Step 2.2.1.2

Raise $- 18$ to the power of $2$ .

$324 {({(2 x - 1)}^{\frac{1}{2}})}^{2} > {(- 36)}^{2}$

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$324 {(2 x - 1)}^{\frac{1}{2} \cdot 2} > {(- 36)}^{2}$

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.

$324 {(2 x - 1)}^{\frac{1}{2} \cdot 2} > {(- 36)}^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$324 {(2 x - 1)}^{1} > {(- 36)}^{2}$

Step 2.2.1.4

Simplify.

$324 (2 x - 1) > {(- 36)}^{2}$

Step 2.2.1.5

Apply the distributive property.

$324 (2 x) + 324 \cdot - 1 > {(- 36)}^{2}$

Step 2.2.1.6

Multiply.

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Step 2.2.1.6.1

Multiply $2$ by $324$ .

$648 x + 324 \cdot - 1 > {(- 36)}^{2}$

Step 2.2.1.6.2

Multiply $324$ by $- 1$ .

$648 x - 324 > {(- 36)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $- 36$ to the power of $2$ .

$648 x - 324 > 1296$

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 $324$ to both sides of the inequality.

$648 x > 1296 + 324$

Step 3.1.2

Add $1296$ and $324$ .

$648 x > 1620$

Step 3.2

Divide each term in $648 x > 1620$ by $648$ and simplify.

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Step 3.2.1

Divide each term in $648 x > 1620$ by $648$ .

$\frac{648 x}{648} > \frac{1620}{648}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $648$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{648 x}{648} > \frac{1620}{648}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{1620}{648}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $1620$ and $648$ .

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Step 3.2.3.1.1

Factor $324$ out of $1620$ .

$x > \frac{324 (5)}{648}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $324$ out of $648$ .

$x > \frac{324 \cdot 5}{324 \cdot 2}$

Step 3.2.3.1.2.2

Cancel the common factor.

$x > \frac{324 \cdot 5}{324 \cdot 2}$

Step 3.2.3.1.2.3

Rewrite the expression.

$x > \frac{5}{2}$

Step 4

Find the domain of $- 18 \sqrt{2 x - 1} + 36$ .

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Step 4.1

Set the radicand in $\sqrt{2 x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$2 x - 1 \geq 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Add $1$ to both sides of the inequality.

$2 x \geq 1$

Step 4.2.2

Divide each term in $2 x \geq 1$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 x \geq 1$ by $2$ .

$\frac{2 x}{2} \geq \frac{1}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{1}{2}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{1}{2}$

Step 4.3

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

$[\frac{1}{2}, \infty)$

Step 5

Use each root to create test intervals.

$x < \frac{1}{2}$

$\frac{1}{2} < x < \frac{5}{2}$

$x > \frac{5}{2}$

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 $x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 6.1.1

Choose a value on the interval $x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 6.1.2

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

$- 18 \sqrt{2 (0) - 1} > - 36$

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 $\frac{1}{2} < x < \frac{5}{2}$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $\frac{1}{2} < x < \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 2$

Step 6.2.2

Replace $x$ with $2$ in the original inequality.

$- 18 \sqrt{2 (2) - 1} > - 36$

Step 6.2.3

The left side $- 31.17691453$ is greater than the right side $- 36$ , which means that the given statement is always true.

True

Step 6.3

Test a value on the interval $x > \frac{5}{2}$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $x > \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 5$

Step 6.3.2

Replace $x$ with $5$ in the original inequality.

$- 18 \sqrt{2 (5) - 1} > - 36$

Step 6.3.3

The left side $- 54$ is not greater than the right side $- 36$ , which means that the given statement is false.

False

Step 6.4

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

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < \frac{5}{2}$ True

$x > \frac{5}{2}$ False

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < \frac{5}{2}$ True

$x > \frac{5}{2}$ False

Step 7

The solution consists of all of the true intervals.

$\frac{1}{2} < x < \frac{5}{2}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$\frac{1}{2} < x < \frac{5}{2}$

Interval Notation:

$(\frac{1}{2}, \frac{5}{2})$

Step 9