Finite Math Examples

$- 5 x^{2} + 22 x + 0.5 < 20$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $20$ from both sides of the inequality.

$- 5 x^{2} + 22 x + 0.5 - 20 < 0$

Step 1.2

Subtract $20$ from $0.5$ .

$- 5 x^{2} + 22 x - 19.5 < 0$

Step 2

Convert the inequality to an equation.

$- 5 x^{2} + 22 x - 19.5 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 5$ , $b = 22$ , and $c = - 19.5$ into the quadratic formula and solve for $x$ .

$\frac{- 22 \pm \sqrt{22^{2} - 4 \cdot (- 5 \cdot - 19.5)}}{2 \cdot - 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $22$ to the power of $2$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot - 5 \cdot - 19.5}}{2 \cdot - 5}$

Step 5.1.2

Multiply $- 4 \cdot - 5 \cdot - 19.5$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 22 \pm \sqrt{484 + 20 \cdot - 19.5}}{2 \cdot - 5}$

Step 5.1.2.2

Multiply $20$ by $- 19.5$ .

$x = \frac{- 22 \pm \sqrt{484 - 390}}{2 \cdot - 5}$

Step 5.1.3

Subtract $390$ from $484$ .

$x = \frac{- 22 \pm \sqrt{94}}{2 \cdot - 5}$

Step 5.2

Multiply $2$ by $- 5$ .

$x = \frac{- 22 \pm \sqrt{94}}{- 10}$

Step 5.3

Simplify $\frac{- 22 \pm \sqrt{94}}{- 10}$ .

$x = \frac{22 \pm \sqrt{94}}{10}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $22$ to the power of $2$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot - 5 \cdot - 19.5}}{2 \cdot - 5}$

Step 6.1.2

Multiply $- 4 \cdot - 5 \cdot - 19.5$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 22 \pm \sqrt{484 + 20 \cdot - 19.5}}{2 \cdot - 5}$

Step 6.1.2.2

Multiply $20$ by $- 19.5$ .

$x = \frac{- 22 \pm \sqrt{484 - 390}}{2 \cdot - 5}$

Step 6.1.3

Subtract $390$ from $484$ .

$x = \frac{- 22 \pm \sqrt{94}}{2 \cdot - 5}$

Step 6.2

Multiply $2$ by $- 5$ .

$x = \frac{- 22 \pm \sqrt{94}}{- 10}$

Step 6.3

Simplify $\frac{- 22 \pm \sqrt{94}}{- 10}$ .

$x = \frac{22 \pm \sqrt{94}}{10}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{22 + \sqrt{94}}{10}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $22$ to the power of $2$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot - 5 \cdot - 19.5}}{2 \cdot - 5}$

Step 7.1.2

Multiply $- 4 \cdot - 5 \cdot - 19.5$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 22 \pm \sqrt{484 + 20 \cdot - 19.5}}{2 \cdot - 5}$

Step 7.1.2.2

Multiply $20$ by $- 19.5$ .

$x = \frac{- 22 \pm \sqrt{484 - 390}}{2 \cdot - 5}$

Step 7.1.3

Subtract $390$ from $484$ .

$x = \frac{- 22 \pm \sqrt{94}}{2 \cdot - 5}$

Step 7.2

Multiply $2$ by $- 5$ .

$x = \frac{- 22 \pm \sqrt{94}}{- 10}$

Step 7.3

Simplify $\frac{- 22 \pm \sqrt{94}}{- 10}$ .

$x = \frac{22 \pm \sqrt{94}}{10}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{22 - \sqrt{94}}{10}$

Step 8

Consolidate the solutions.

$x = \frac{22 + \sqrt{94}}{10}, \frac{22 - \sqrt{94}}{10}$

Step 9

Use each root to create test intervals.

$x < \frac{22 - \sqrt{94}}{10}$

$\frac{22 - \sqrt{94}}{10} < x < \frac{22 + \sqrt{94}}{10}$

$x > \frac{22 + \sqrt{94}}{10}$

Step 10

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 10.1

Test a value on the interval $x < \frac{22 - \sqrt{94}}{10}$ to see if it makes the inequality true.

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Step 10.1.1

Choose a value on the interval $x < \frac{22 - \sqrt{94}}{10}$ and see if this value makes the original inequality true.

$x = 0$

Step 10.1.2

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

$- 5 {(0)}^{2} + 22 (0) + 0.5 < 20$

Step 10.1.3

The left side $0.5$ is less than the right side $20$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $\frac{22 - \sqrt{94}}{10} < x < \frac{22 + \sqrt{94}}{10}$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $\frac{22 - \sqrt{94}}{10} < x < \frac{22 + \sqrt{94}}{10}$ and see if this value makes the original inequality true.

$x = 2$

Step 10.2.2

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

$- 5 {(2)}^{2} + 22 (2) + 0.5 < 20$

Step 10.2.3

The left side $24.5$ is not less than the right side $20$ , which means that the given statement is false.

False

Step 10.3

Test a value on the interval $x > \frac{22 + \sqrt{94}}{10}$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $x > \frac{22 + \sqrt{94}}{10}$ and see if this value makes the original inequality true.

$x = 6$

Step 10.3.2

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

$- 5 {(6)}^{2} + 22 (6) + 0.5 < 20$

Step 10.3.3

The left side $- 47.5$ is less than the right side $20$ , which means that the given statement is always true.

True

Step 10.4

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

$x < \frac{22 - \sqrt{94}}{10}$ True

$\frac{22 - \sqrt{94}}{10} < x < \frac{22 + \sqrt{94}}{10}$ False

$x > \frac{22 + \sqrt{94}}{10}$ True

$x < \frac{22 - \sqrt{94}}{10}$ True

$\frac{22 - \sqrt{94}}{10} < x < \frac{22 + \sqrt{94}}{10}$ False

$x > \frac{22 + \sqrt{94}}{10}$ True

Step 11

The solution consists of all of the true intervals.

$x < \frac{22 - \sqrt{94}}{10}$ or $x > \frac{22 + \sqrt{94}}{10}$

Step 12

Convert the inequality to interval notation.

$(- \infty, \frac{22 - \sqrt{94}}{10}) \cup (\frac{22 + \sqrt{94}}{10}, \infty)$

Step 13