Finite Math Examples

$\frac{5 t}{t^{2} + 1} + 98.6 > 100$

Step 1

Subtract $100$ from both sides of the inequality.

$\frac{5 t}{t^{2} + 1} + 98.6 - 100 > 0$

Step 2

Simplify $\frac{5 t}{t^{2} + 1} + 98.6 - 100$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Write $98.6$ as a fraction with denominator $1$ .

$\frac{5 t}{t^{2} + 1} + \frac{98.6}{1} - 100 > 0$

Step 2.1.2

Multiply $\frac{98.6}{1}$ by $\frac{t^{2} + 1}{t^{2} + 1}$ .

$\frac{5 t}{t^{2} + 1} + \frac{98.6}{1} \cdot \frac{t^{2} + 1}{t^{2} + 1} - 100 > 0$

Step 2.1.3

Multiply $\frac{98.6}{1}$ by $\frac{t^{2} + 1}{t^{2} + 1}$ .

$\frac{5 t}{t^{2} + 1} + \frac{98.6 (t^{2} + 1)}{t^{2} + 1} - 100 > 0$

Step 2.1.4

Write $- 100$ as a fraction with denominator $1$ .

$\frac{5 t}{t^{2} + 1} + \frac{98.6 (t^{2} + 1)}{t^{2} + 1} + \frac{- 100}{1} > 0$

Step 2.1.5

Multiply $\frac{- 100}{1}$ by $\frac{t^{2} + 1}{t^{2} + 1}$ .

$\frac{5 t}{t^{2} + 1} + \frac{98.6 (t^{2} + 1)}{t^{2} + 1} + \frac{- 100}{1} \cdot \frac{t^{2} + 1}{t^{2} + 1} > 0$

Step 2.1.6

Multiply $\frac{- 100}{1}$ by $\frac{t^{2} + 1}{t^{2} + 1}$ .

$\frac{5 t}{t^{2} + 1} + \frac{98.6 (t^{2} + 1)}{t^{2} + 1} + \frac{- 100 (t^{2} + 1)}{t^{2} + 1} > 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{5 t + 98.6 (t^{2} + 1) - 100 (t^{2} + 1)}{t^{2} + 1} > 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{5 t + 98.6 t^{2} + 98.6 \cdot 1 - 100 (t^{2} + 1)}{t^{2} + 1} > 0$

Step 2.3.2

Multiply $98.6$ by $1$ .

$\frac{5 t + 98.6 t^{2} + 98.6 - 100 (t^{2} + 1)}{t^{2} + 1} > 0$

Step 2.3.3

Apply the distributive property.

$\frac{5 t + 98.6 t^{2} + 98.6 - 100 t^{2} - 100 \cdot 1}{t^{2} + 1} > 0$

Step 2.3.4

Multiply $- 100$ by $1$ .

$\frac{5 t + 98.6 t^{2} + 98.6 - 100 t^{2} - 100}{t^{2} + 1} > 0$

Step 2.4

Subtract $100 t^{2}$ from $98.6 t^{2}$ .

$\frac{5 t - 1.4 t^{2} + 98.6 - 100}{t^{2} + 1} > 0$

Step 2.5

Subtract $100$ from $98.6$ .

$\frac{5 t - 1.4 t^{2} - 1.4}{t^{2} + 1} > 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Factor $0.2$ out of $5 t - 1.4 t^{2} - 1.4$ .

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Step 2.6.1.1

Factor $0.2$ out of $5 t$ .

$\frac{0.2 (25 t) - 1.4 t^{2} - 1.4}{t^{2} + 1} > 0$

Step 2.6.1.2

Factor $0.2$ out of $- 1.4 t^{2}$ .

$\frac{0.2 (25 t) + 0.2 (- 7 t^{2}) - 1.4}{t^{2} + 1} > 0$

Step 2.6.1.3

Factor $0.2$ out of $- 1.4$ .

$\frac{0.2 (25 t) + 0.2 (- 7 t^{2}) + 0.2 (- 7)}{t^{2} + 1} > 0$

Step 2.6.1.4

Factor $0.2$ out of $0.2 (25 t) + 0.2 (- 7 t^{2})$ .

$\frac{0.2 (25 t - 7 t^{2}) + 0.2 (- 7)}{t^{2} + 1} > 0$

Step 2.6.1.5

Factor $0.2$ out of $0.2 (25 t - 7 t^{2}) + 0.2 (- 7)$ .

$\frac{0.2 (25 t - 7 t^{2} - 7)}{t^{2} + 1} > 0$

Step 2.6.2

Reorder terms.

$\frac{0.2 (- 7 t^{2} + 25 t - 7)}{t^{2} + 1} > 0$

Step 2.7

Factor $- 1$ out of $- 7 t^{2}$ .

$\frac{0.2 (- (7 t^{2}) + 25 t - 7)}{t^{2} + 1} > 0$

Step 2.8

Factor $- 1$ out of $25 t$ .

$\frac{0.2 (- (7 t^{2}) - (- 25 t) - 7)}{t^{2} + 1} > 0$

Step 2.9

Factor $- 1$ out of $- (7 t^{2}) - (- 25 t)$ .

$\frac{0.2 (- (7 t^{2} - 25 t) - 7)}{t^{2} + 1} > 0$

Step 2.10

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{0.2 (- (7 t^{2} - 25 t) - 1 (7))}{t^{2} + 1} > 0$

Step 2.11

Factor $- 1$ out of $- (7 t^{2} - 25 t) - 1 (7)$ .

$\frac{0.2 (- (7 t^{2} - 25 t + 7))}{t^{2} + 1} > 0$

Step 2.12

Rewrite $- (7 t^{2} - 25 t + 7)$ as $- 1 (7 t^{2} - 25 t + 7)$ .

$\frac{0.2 (- 1 (7 t^{2} - 25 t + 7))}{t^{2} + 1} > 0$

Step 2.13

Move the negative in front of the fraction.

$- \frac{0.2 (7 t^{2} - 25 t + 7)}{t^{2} + 1} > 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$7 t^{2} - 25 t + 7 = 0$

$t^{2} + 1 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 7$ , $b = - 25$ , and $c = 7$ into the quadratic formula and solve for $t$ .

$\frac{25 \pm \sqrt{{(- 25)}^{2} - 4 \cdot (7 \cdot 7)}}{2 \cdot 7}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 25$ to the power of $2$ .

$t = \frac{25 \pm \sqrt{625 - 4 \cdot 7 \cdot 7}}{2 \cdot 7}$

Step 6.1.2

Multiply $- 4 \cdot 7 \cdot 7$ .

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Step 6.1.2.1

Multiply $- 4$ by $7$ .

$t = \frac{25 \pm \sqrt{625 - 28 \cdot 7}}{2 \cdot 7}$

Step 6.1.2.2

Multiply $- 28$ by $7$ .

$t = \frac{25 \pm \sqrt{625 - 196}}{2 \cdot 7}$

Step 6.1.3

Subtract $196$ from $625$ .

$t = \frac{25 \pm \sqrt{429}}{2 \cdot 7}$

Step 6.2

Multiply $2$ by $7$ .

$t = \frac{25 \pm \sqrt{429}}{14}$

Step 7

The final answer is the combination of both solutions.

$t = \frac{25 + \sqrt{429}}{14}, \frac{25 - \sqrt{429}}{14}$

Step 8

Subtract $1$ from both sides of the equation.

$t^{2} = - 1$

Step 9

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{- 1}$

Step 10

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \pm i$

Step 11

The complete solution is the result of both the positive and negative portions of the solution.

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Step 11.1

First, use the positive value of the $\pm$ to find the first solution.

$t = i$

Step 11.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - i$

Step 11.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = i, - i$

Step 12

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$t = \frac{25 + \sqrt{429}}{14}, \frac{25 - \sqrt{429}}{14}$

$t = i, - i$

Step 13

Consolidate the solutions.

$t = \frac{25 + \sqrt{429}}{14}, \frac{25 - \sqrt{429}}{14}$

Step 14

Use each root to create test intervals.

$t < \frac{25 - \sqrt{429}}{14}$

$\frac{25 - \sqrt{429}}{14} < t < \frac{25 + \sqrt{429}}{14}$

$t > \frac{25 + \sqrt{429}}{14}$

Step 15

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 15.1

Test a value on the interval $t < \frac{25 - \sqrt{429}}{14}$ to see if it makes the inequality true.

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Step 15.1.1

Choose a value on the interval $t < \frac{25 - \sqrt{429}}{14}$ and see if this value makes the original inequality true.

$t = 0$

Step 15.1.2

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

$\frac{5 (0)}{{(0)}^{2} + 1} + 98.6 > 100$

Step 15.1.3

The left side $98.6$ is not greater than the right side $100$ , which means that the given statement is false.

False

Step 15.2

Test a value on the interval $\frac{25 - \sqrt{429}}{14} < t < \frac{25 + \sqrt{429}}{14}$ to see if it makes the inequality true.

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Step 15.2.1

Choose a value on the interval $\frac{25 - \sqrt{429}}{14} < t < \frac{25 + \sqrt{429}}{14}$ and see if this value makes the original inequality true.

$t = 2$

Step 15.2.2

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

$\frac{5 (2)}{{(2)}^{2} + 1} + 98.6 > 100$

Step 15.2.3

The left side $100.6$ is greater than the right side $100$ , which means that the given statement is always true.

True

Step 15.3

Test a value on the interval $t > \frac{25 + \sqrt{429}}{14}$ to see if it makes the inequality true.

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Step 15.3.1

Choose a value on the interval $t > \frac{25 + \sqrt{429}}{14}$ and see if this value makes the original inequality true.

$t = 6$

Step 15.3.2

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

$\frac{5 (6)}{{(6)}^{2} + 1} + 98.6 > 100$

Step 15.3.3

The left side $99.4\overline{108}$ is not greater than the right side $100$ , which means that the given statement is false.

False

Step 15.4

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

$t < \frac{25 - \sqrt{429}}{14}$ False

$\frac{25 - \sqrt{429}}{14} < t < \frac{25 + \sqrt{429}}{14}$ True

$t > \frac{25 + \sqrt{429}}{14}$ False

$t < \frac{25 - \sqrt{429}}{14}$ False

$\frac{25 - \sqrt{429}}{14} < t < \frac{25 + \sqrt{429}}{14}$ True

$t > \frac{25 + \sqrt{429}}{14}$ False

Step 16

The solution consists of all of the true intervals.

$\frac{25 - \sqrt{429}}{14} < t < \frac{25 + \sqrt{429}}{14}$

Step 17

The result can be shown in multiple forms.

Inequality Form:

$\frac{25 - \sqrt{429}}{14} < t < \frac{25 + \sqrt{429}}{14}$

Interval Notation:

$(\frac{25 - \sqrt{429}}{14}, \frac{25 + \sqrt{429}}{14})$

Step 18