Finite Math Examples

$25 x^{3} + 75 x^{2} - 36 x - 108 > 0$

Step 1

Convert the inequality to an equation.

$25 x^{3} + 75 x^{2} - 36 x - 108 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

$(25 x^{3} + 75 x^{2}) - 36 x - 108 = 0$

Step 2.1.2

Factor out the greatest common factor (GCF) from each group.

$25 x^{2} (x + 3) - 36 (x + 3) = 0$

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $x + 3$ .

$(x + 3) (25 x^{2} - 36) = 0$

Step 2.3

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$(x + 3) ({(5 x)}^{2} - 36) = 0$

Step 2.4

Rewrite $36$ as $6^{2}$ .

$(x + 3) ({(5 x)}^{2} - 6^{2}) = 0$

Step 2.5

Factor.

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Step 2.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x$ and $b = 6$ .

$(x + 3) ((5 x + 6) (5 x - 6)) = 0$

Step 2.5.2

Remove unnecessary parentheses.

$(x + 3) (5 x + 6) (5 x - 6) = 0$

Step 3

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

$x + 3 = 0$

$5 x + 6 = 0$

$5 x - 6 = 0$

Step 4

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5

Set $5 x + 6$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $5 x + 6$ equal to $0$ .

$5 x + 6 = 0$

Step 5.2

Solve $5 x + 6 = 0$ for $x$ .

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Step 5.2.1

Subtract $6$ from both sides of the equation.

$5 x = - 6$

Step 5.2.2

Divide each term in $5 x = - 6$ by $5$ and simplify.

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Step 5.2.2.1

Divide each term in $5 x = - 6$ by $5$ .

$\frac{5 x}{5} = \frac{- 6}{5}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 6}{5}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{5}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{6}{5}$

Step 6

Set $5 x - 6$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $5 x - 6$ equal to $0$ .

$5 x - 6 = 0$

Step 6.2

Solve $5 x - 6 = 0$ for $x$ .

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Step 6.2.1

Add $6$ to both sides of the equation.

$5 x = 6$

Step 6.2.2

Divide each term in $5 x = 6$ by $5$ and simplify.

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Step 6.2.2.1

Divide each term in $5 x = 6$ by $5$ .

$\frac{5 x}{5} = \frac{6}{5}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $5$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{6}{5}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{5}$

Step 7

The final solution is all the values that make $(x + 3) (5 x + 6) (5 x - 6) = 0$ true.

$x = - 3, - \frac{6}{5}, \frac{6}{5}$

Step 8

Use each root to create test intervals.

$x < - 3$

$- 3 < x < - \frac{6}{5}$

$- \frac{6}{5} < x < \frac{6}{5}$

$x > \frac{6}{5}$

Step 9

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 9.1

Test a value on the interval $x < - 3$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

Step 9.1.2

Replace $x$ with $- 6$ in the original inequality.

$25 {(- 6)}^{3} + 75 {(- 6)}^{2} - 36 \cdot - 6 - 108 > 0$

Step 9.1.3

The left side $- 2592$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- 3 < x < - \frac{6}{5}$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- 3 < x < - \frac{6}{5}$ and see if this value makes the original inequality true.

$x = - 2$

Step 9.2.2

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

$25 {(- 2)}^{3} + 75 {(- 2)}^{2} - 36 \cdot - 2 - 108 > 0$

Step 9.2.3

The left side $64$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $- \frac{6}{5} < x < \frac{6}{5}$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $- \frac{6}{5} < x < \frac{6}{5}$ and see if this value makes the original inequality true.

$x = 0$

Step 9.3.2

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

$25 {(0)}^{3} + 75 {(0)}^{2} - 36 \cdot 0 - 108 > 0$

Step 9.3.3

The left side $- 108$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 9.4

Test a value on the interval $x > \frac{6}{5}$ to see if it makes the inequality true.

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Step 9.4.1

Choose a value on the interval $x > \frac{6}{5}$ and see if this value makes the original inequality true.

$x = 4$

Step 9.4.2

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

$25 {(4)}^{3} + 75 {(4)}^{2} - 36 \cdot 4 - 108 > 0$

Step 9.4.3

The left side $2548$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.5

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

$x < - 3$ False

$- 3 < x < - \frac{6}{5}$ True

$- \frac{6}{5} < x < \frac{6}{5}$ False

$x > \frac{6}{5}$ True

$x < - 3$ False

$- 3 < x < - \frac{6}{5}$ True

$- \frac{6}{5} < x < \frac{6}{5}$ False

$x > \frac{6}{5}$ True

Step 10

The solution consists of all of the true intervals.

$- 3 < x < - \frac{6}{5}$ or $x > \frac{6}{5}$

Step 11

Convert the inequality to interval notation.

$(- 3, - \frac{6}{5}) \cup (\frac{6}{5}, \infty)$

Step 12