Calculus Examples

$0.01 x^{2} + x - 600 > 0$

Step 1

Convert the inequality to an equation.

$0.01 x^{2} + x - 600 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 0.01$ , $b = 1$ , and $c = - 600$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (0.01 \cdot - 600)}}{2 \cdot 0.01}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 0.01 \cdot - 600}}{2 \cdot 0.01}$

Step 4.1.2

Multiply $- 4 \cdot 0.01 \cdot - 600$ .

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Step 4.1.2.1

Multiply $- 4$ by $0.01$ .

$x = \frac{- 1 \pm \sqrt{1 - 0.04 \cdot - 600}}{2 \cdot 0.01}$

Step 4.1.2.2

Multiply $- 0.04$ by $- 600$ .

$x = \frac{- 1 \pm \sqrt{1 + 24}}{2 \cdot 0.01}$

Step 4.1.3

Add $1$ and $24$ .

$x = \frac{- 1 \pm \sqrt{25}}{2 \cdot 0.01}$

Step 4.1.4

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 1 \pm \sqrt{5^{2}}}{2 \cdot 0.01}$

Step 4.1.5

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

$x = \frac{- 1 \pm 5}{2 \cdot 0.01}$

Step 4.2

Multiply $2$ by $0.01$ .

$x = \frac{- 1 \pm 5}{0.02}$

Step 4.3

Simplify $\frac{- 1 \pm 5}{0.02}$ .

$x = - 50 \pm 250$

Step 5

Consolidate the solutions.

$x = 200, - 300$

Step 6

Use each root to create test intervals.

$x < - 300$

$- 300 < x < 200$

$x > 200$

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 $x < - 300$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - 300$ and see if this value makes the original inequality true.

$x = - 302$

Step 7.1.2

Replace $x$ with $- 302$ in the original inequality.

$0.01 {(- 302)}^{2} - 302 - 600 > 0$

Step 7.1.3

The left side $10.04$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $- 300 < x < 200$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 300 < x < 200$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

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

$0.01 {(0)}^{2} + 0 - 600 > 0$

Step 7.2.3

The left side $- 600$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.3

Test a value on the interval $x > 200$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > 200$ and see if this value makes the original inequality true.

$x = 202$

Step 7.3.2

Replace $x$ with $202$ in the original inequality.

$0.01 {(202)}^{2} + 202 - 600 > 0$

Step 7.3.3

The left side $10.04$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.4

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

$x < - 300$ True

$- 300 < x < 200$ False

$x > 200$ True

$x < - 300$ True

$- 300 < x < 200$ False

$x > 200$ True

Step 8

The solution consists of all of the true intervals.

$x < - 300$ or $x > 200$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x < - 300 or x > 200$

Interval Notation:

$(- \infty, - 300) \cup (200, \infty)$

Step 10