Finite Math Examples

$10000000 + 400000 n < 8000000 \cdot {1.028}^{n}$

Step 1

Graph each side of the equation. The solution is the x-value of the point of intersection.

$n \approx - 9.69967345, 45.75195616$

Step 2

Use each root to create test intervals.

$n < - 9.69967345$

$- 9.69967345 < n < 45.75195616$

$n > 45.75195616$

Step 3

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 3.1

Test a value on the interval $n < - 9.69967345$ to see if it makes the inequality true.

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Step 3.1.1

Choose a value on the interval $n < - 9.69967345$ and see if this value makes the original inequality true.

$n = - 12$

Step 3.1.2

Replace $n$ with $- 12$ in the original inequality.

$10000000 + 400000 (- 12) < 8000000 \cdot {1.028}^{- 12}$

Step 3.1.3

The left side $5200000$ is less than the right side $5743446.9112417$ , which means that the given statement is always true.

True

Step 3.2

Test a value on the interval $- 9.69967345 < n < 45.75195616$ to see if it makes the inequality true.

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Step 3.2.1

Choose a value on the interval $- 9.69967345 < n < 45.75195616$ and see if this value makes the original inequality true.

$n = 0$

Step 3.2.2

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

$10000000 + 400000 (0) < 8000000 \cdot {1.028}^{0}$

Step 3.2.3

The left side $10000000$ is not less than the right side $8000000$ , which means that the given statement is false.

False

Step 3.3

Test a value on the interval $n > 45.75195616$ to see if it makes the inequality true.

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Step 3.3.1

Choose a value on the interval $n > 45.75195616$ and see if this value makes the original inequality true.

$n = 48$

Step 3.3.2

Replace $n$ with $48$ in the original inequality.

$10000000 + 400000 (48) < 8000000 \cdot {1.028}^{48}$

Step 3.3.3

The left side $29200000$ is less than the right side $30113379.0365008$ , which means that the given statement is always true.

True

Step 3.4

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

$n < - 9.69967345$ True

$- 9.69967345 < n < 45.75195616$ False

$n > 45.75195616$ True

$n < - 9.69967345$ True

$- 9.69967345 < n < 45.75195616$ False

$n > 45.75195616$ True

Step 4

The solution consists of all of the true intervals.

$n < - 9.69967345$ or $n > 45.75195616$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 9.69967345) \cup (45.75195616, \infty)$

Step 6