Finite Math Examples

$x^{2} + 2 x - 60 > 3$

Step 1

Convert the inequality to an equation.

$x^{2} + 2 x - 60 = 3$

Step 2

Subtract $3$ from both sides of the equation.

$x^{2} + 2 x - 60 - 3 = 0$

Step 3

Subtract $3$ from $- 60$ .

$x^{2} + 2 x - 63 = 0$

Step 4

Factor $x^{2} + 2 x - 63$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 63$ and whose sum is $2$ .

$- 7, 9$

Step 4.2

Write the factored form using these integers.

$(x - 7) (x + 9) = 0$

Step 5

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

$x - 7 = 0$

$x + 9 = 0$

Step 6

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 7

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 7.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 8

The final solution is all the values that make $(x - 7) (x + 9) = 0$ true.

$x = 7, - 9$

Step 9

Use each root to create test intervals.

$x < - 9$

$- 9 < x < 7$

$x > 7$

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 $x < - 9$ to see if it makes the inequality true.

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Step 10.1.1

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Step 10.1.2

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

${(- 12)}^{2} + 2 (- 12) - 60 > 3$

Step 10.1.3

The left side $60$ is greater than the right side $3$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $- 9 < x < 7$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- 9 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

Step 10.2.2

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

${(0)}^{2} + 2 (0) - 60 > 3$

Step 10.2.3

The left side $- 60$ is not greater than the right side $3$ , which means that the given statement is false.

False

Step 10.3

Test a value on the interval $x > 7$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 10.3.2

Replace $x$ with $10$ in the original inequality.

${(10)}^{2} + 2 (10) - 60 > 3$

Step 10.3.3

The left side $60$ is greater than the right side $3$ , which means that the given statement is always true.

True

Step 10.4

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

$x < - 9$ True

$- 9 < x < 7$ False

$x > 7$ True

$x < - 9$ True

$- 9 < x < 7$ False

$x > 7$ True

Step 11

The solution consists of all of the true intervals.

$x < - 9$ or $x > 7$

Step 12

Convert the inequality to interval notation.

$(- \infty, - 9) \cup (7, \infty)$

Step 13