Precalculus Examples

${(2 x - 1)}^{\frac{2}{3}} < x^{\frac{1}{3}}$

Step 1

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{1}{4}, 1$

Step 2

Use each root to create test intervals.

$x < \frac{1}{4}$

$\frac{1}{4} < x < 1$

$x > 1$

Step 3

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 3.1

Test a value on the interval $x < \frac{1}{4}$ to see if it makes the inequality true.

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Step 3.1.1

Choose a value on the interval $x < \frac{1}{4}$ and see if this value makes the original inequality true.

$x = 0$

Step 3.1.2

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

${(2 (0) - 1)}^{\frac{2}{3}} < {(0)}^{\frac{1}{3}}$

Step 3.1.3

The left side $1$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 3.2

Test a value on the interval $\frac{1}{4} < x < 1$ to see if it makes the inequality true.

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Step 3.2.1

Choose a value on the interval $\frac{1}{4} < x < 1$ and see if this value makes the original inequality true.

$x = 0.62$

Step 3.2.2

Replace $x$ with $0.62$ in the original inequality.

${(2 (0.62) - 1)}^{\frac{2}{3}} < {(0.62)}^{\frac{1}{3}}$

Step 3.2.3

The left side $0.38619575$ is less than the right side $0.85270189$ , which means that the given statement is always true.

True

Step 3.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 3.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 3.3.2

Replace $x$ with $4$ in the original inequality.

${(2 (4) - 1)}^{\frac{2}{3}} < {(4)}^{\frac{1}{3}}$

Step 3.3.3

The left side $3.65930571$ is not less than the right side $1.58740105$ , which means that the given statement is false.

False

Step 3.4

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

$x < \frac{1}{4}$ False

$\frac{1}{4} < x < 1$ True

$x > 1$ False

$x < \frac{1}{4}$ False

$\frac{1}{4} < x < 1$ True

$x > 1$ False

Step 4

The solution consists of all of the true intervals.

$\frac{1}{4} < x < 1$

Step 5

Convert the inequality to interval notation.

$(\frac{1}{4}, 1)$

Step 6