Precalculus Examples

$x^{4} (1 - 3 x) {(5 x + 2)}^{3} > 0$

Step 1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{4} = 0$

$1 - 3 x = 0$

${(5 x + 2)}^{3} = 0$

Step 2

Set $x^{4}$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $x^{4}$ equal to $0$ .

$x^{4} = 0$

Step 2.2

Solve $x^{4} = 0$ for $x$ .

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Step 2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{0}$

Step 2.2.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 2.2.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Set $1 - 3 x$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $1 - 3 x$ equal to $0$ .

$1 - 3 x = 0$

Step 3.2

Solve $1 - 3 x = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$- 3 x = - 1$

Step 3.2.2

Divide each term in $- 3 x = - 1$ by $- 3$ and simplify.

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Step 3.2.2.1

Divide each term in $- 3 x = - 1$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 1}{- 3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 1}{- 3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 3}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{3}$

Step 4

Set ${(5 x + 2)}^{3}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set ${(5 x + 2)}^{3}$ equal to $0$ .

${(5 x + 2)}^{3} = 0$

Step 4.2

Solve ${(5 x + 2)}^{3} = 0$ for $x$ .

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Step 4.2.1

Set the $5 x + 2$ equal to $0$ .

$5 x + 2 = 0$

Step 4.2.2

Solve for $x$ .

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Step 4.2.2.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 4.2.2.2

Divide each term in $5 x = - 2$ by $5$ and simplify.

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Step 4.2.2.2.1

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Step 4.2.2.2.3

Simplify the right side.

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Step 4.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 5

The final solution is all the values that make $x^{4} (1 - 3 x) {(5 x + 2)}^{3} > 0$ true.

$x = 0, \frac{1}{3}, - \frac{2}{5}$

Step 6

Use each root to create test intervals.

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

$- \frac{2}{5} < x < 0$

$0 < x < \frac{1}{3}$

$x > \frac{1}{3}$

Step 7

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 7.1

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

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Step 7.1.1

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

$x = - 3$

Step 7.1.2

Replace $x$ with $- 3$ in the original inequality.

${(- 3)}^{4} (1 - 3 \cdot - 3) {(5 (- 3) + 2)}^{3} > 0$

Step 7.1.3

The left side $- 1779570$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- \frac{2}{5} < x < 0$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- \frac{2}{5} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.2$

Step 7.2.2

Replace $x$ with $- 0.2$ in the original inequality.

${(- 0.2)}^{4} (1 - 3 \cdot - 0.2) {(5 (- 0.2) + 2)}^{3} > 0$

Step 7.2.3

The left side $0.00256$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $0 < x < \frac{1}{3}$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $0 < x < \frac{1}{3}$ and see if this value makes the original inequality true.

$x = 0.17$

Step 7.3.2

Replace $x$ with $0.17$ in the original inequality.

${(0.17)}^{4} (1 - 3 \cdot 0.17) {(5 (0.17) + 2)}^{3} > 0$

Step 7.3.3

The left side $0.00947384$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.4

Test a value on the interval $x > \frac{1}{3}$ to see if it makes the inequality true.

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Step 7.4.1

Choose a value on the interval $x > \frac{1}{3}$ and see if this value makes the original inequality true.

$x = 3$

Step 7.4.2

Replace $x$ with $3$ in the original inequality.

${(3)}^{4} (1 - 3 \cdot 3) {(5 (3) + 2)}^{3} > 0$

Step 7.4.3

The left side $- 3183624$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.5

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

$x < - \frac{2}{5}$ False

$- \frac{2}{5} < x < 0$ True

$0 < x < \frac{1}{3}$ True

$x > \frac{1}{3}$ False

$x < - \frac{2}{5}$ False

$- \frac{2}{5} < x < 0$ True

$0 < x < \frac{1}{3}$ True

$x > \frac{1}{3}$ False

Step 8

The solution consists of all of the true intervals.

$- \frac{2}{5} < x < 0$ or $0 < x < \frac{1}{3}$

Step 9

Convert the inequality to interval notation.

$(- \frac{2}{5}, 0) \cup (0, \frac{1}{3})$

Step 10