Finite Math Examples

$- 5 p (p - 3000 p + 250000) > 0$

Step 1

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

$p = 0$

$p - 3000 p + 250000 = 0$

Step 2

Set $p$ equal to $0$ .

$p = 0$

Step 3

Set $p - 3000 p + 250000$ equal to $0$ and solve for $p$ .

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Step 3.1

Set $p - 3000 p + 250000$ equal to $0$ .

$p - 3000 p + 250000 = 0$

Step 3.2

Solve $p - 3000 p + 250000 = 0$ for $p$ .

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Step 3.2.1

Subtract $3000 p$ from $p$ .

$- 2999 p + 250000 = 0$

Step 3.2.2

Subtract $250000$ from both sides of the equation.

$- 2999 p = - 250000$

Step 3.2.3

Divide each term in $- 2999 p = - 250000$ by $- 2999$ and simplify.

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Step 3.2.3.1

Divide each term in $- 2999 p = - 250000$ by $- 2999$ .

$\frac{- 2999 p}{- 2999} = \frac{- 250000}{- 2999}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $- 2999$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

$\frac{- 2999 p}{- 2999} = \frac{- 250000}{- 2999}$

Step 3.2.3.2.1.2

Divide $p$ by $1$ .

$p = \frac{- 250000}{- 2999}$

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

Dividing two negative values results in a positive value.

$p = \frac{250000}{2999}$

Step 4

The final solution is all the values that make $- 5 p (p - 3000 p + 250000) > 0$ true.

$p = 0, \frac{250000}{2999}$

Step 5

Use each root to create test intervals.

$p < 0$

$0 < p < \frac{250000}{2999}$

$p > \frac{250000}{2999}$

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 $p < 0$ to see if it makes the inequality true.

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Step 6.1.1

Choose a value on the interval $p < 0$ and see if this value makes the original inequality true.

$p = - 2$

Step 6.1.2

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

$- 5 \cdot - 2 ((- 2) - 3000 \cdot - 2 + 250000) > 0$

Step 6.1.3

The left side $2559980$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.2

Test a value on the interval $0 < p < \frac{250000}{2999}$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $0 < p < \frac{250000}{2999}$ and see if this value makes the original inequality true.

$p = 2$

Step 6.2.2

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

$- 5 \cdot 2 ((2) - 3000 \cdot 2 + 250000) > 0$

Step 6.2.3

The left side $- 2440020$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 6.3

Test a value on the interval $p > \frac{250000}{2999}$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $p > \frac{250000}{2999}$ and see if this value makes the original inequality true.

$p = 86$

Step 6.3.2

Replace $p$ with $86$ in the original inequality.

$- 5 \cdot 86 ((86) - 3000 \cdot 86 + 250000) > 0$

Step 6.3.3

The left side $3403020$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.4

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

$p < 0$ True

$0 < p < \frac{250000}{2999}$ False

$p > \frac{250000}{2999}$ True

$p < 0$ True

$0 < p < \frac{250000}{2999}$ False

$p > \frac{250000}{2999}$ True

Step 7

The solution consists of all of the true intervals.

$p < 0$ or $p > \frac{250000}{2999}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$p < 0 or p > \frac{250000}{2999}$

Interval Notation:

$(- \infty, 0) \cup (\frac{250000}{2999}, \infty)$

Step 9