Trigonometry Examples

$x^{3} + 6 x^{2} > - x^{2} + 7 x + 5$

Step 1

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

$x \approx - 7.81394463, - 0.49052873, 1.30447337$

Step 2

Use each root to create test intervals.

$x < - 7.81394463$

$- 7.81394463 < x < - 0.49052873$

$- 0.49052873 < x < 1.30447337$

$x > 1.30447337$

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 $x < - 7.81394463$ to see if it makes the inequality true.

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Step 3.1.1

Choose a value on the interval $x < - 7.81394463$ and see if this value makes the original inequality true.

$x = - 10$

Step 3.1.2

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

${(- 10)}^{3} + 6 {(- 10)}^{2} > - {(- 10)}^{2} + 7 (- 10) + 5$

Step 3.1.3

The left side $- 400$ is not greater than the right side $- 165$ , which means that the given statement is false.

False

Step 3.2

Test a value on the interval $- 7.81394463 < x < - 0.49052873$ to see if it makes the inequality true.

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Step 3.2.1

Choose a value on the interval $- 7.81394463 < x < - 0.49052873$ and see if this value makes the original inequality true.

$x = - 4$

Step 3.2.2

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

${(- 4)}^{3} + 6 {(- 4)}^{2} > - {(- 4)}^{2} + 7 (- 4) + 5$

Step 3.2.3

The left side $32$ is greater than the right side $- 39$ , which means that the given statement is always true.

True

Step 3.3

Test a value on the interval $- 0.49052873 < x < 1.30447337$ to see if it makes the inequality true.

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Step 3.3.1

Choose a value on the interval $- 0.49052873 < x < 1.30447337$ and see if this value makes the original inequality true.

$x = 0$

Step 3.3.2

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

${(0)}^{3} + 6 {(0)}^{2} > - {(0)}^{2} + 7 (0) + 5$

Step 3.3.3

The left side $0$ is not greater than the right side $5$ , which means that the given statement is false.

False

Step 3.4

Test a value on the interval $x > 1.30447337$ to see if it makes the inequality true.

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Step 3.4.1

Choose a value on the interval $x > 1.30447337$ and see if this value makes the original inequality true.

$x = 4$

Step 3.4.2

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

${(4)}^{3} + 6 {(4)}^{2} > - {(4)}^{2} + 7 (4) + 5$

Step 3.4.3

The left side $160$ is greater than the right side $17$ , which means that the given statement is always true.

True

Step 3.5

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

$x < - 7.81394463$ False

$- 7.81394463 < x < - 0.49052873$ True

$- 0.49052873 < x < 1.30447337$ False

$x > 1.30447337$ True

$x < - 7.81394463$ False

$- 7.81394463 < x < - 0.49052873$ True

$- 0.49052873 < x < 1.30447337$ False

$x > 1.30447337$ True

Step 4

The solution consists of all of the true intervals.

$- 7.81394463 < x < - 0.49052873$ or $x > 1.30447337$

Step 5

Convert the inequality to interval notation.

$(- 7.81394463, - 0.49052873) \cup (1.30447337, \infty)$

Step 6