Pre-Algebra Examples

$r^{2} > 3 r + 10$

Step 1

Subtract $3 r$ from both sides of the inequality.

$r^{2} - 3 r > 10$

Step 2

Convert the inequality to an equation.

$r^{2} - 3 r = 10$

Step 3

Subtract $10$ from both sides of the equation.

$r^{2} - 3 r - 10 = 0$

Step 4

Factor $r^{2} - 3 r - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 4.2

Write the factored form using these integers.

$(r - 5) (r + 2) = 0$

Step 5

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

$r - 5 = 0$

$r + 2 = 0$

Step 6

Set $r - 5$ equal to $0$ and solve for $r$ .

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Step 6.1

Set $r - 5$ equal to $0$ .

$r - 5 = 0$

Step 6.2

Add $5$ to both sides of the equation.

$r = 5$

Step 7

Set $r + 2$ equal to $0$ and solve for $r$ .

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Step 7.1

Set $r + 2$ equal to $0$ .

$r + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$r = - 2$

Step 8

The final solution is all the values that make $(r - 5) (r + 2) = 0$ true.

$r = 5, - 2$

Step 9

Use each root to create test intervals.

$r < - 2$

$- 2 < r < 5$

$r > 5$

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 $r < - 2$ to see if it makes the inequality true.

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Step 10.1.1

Choose a value on the interval $r < - 2$ and see if this value makes the original inequality true.

$r = - 4$

Step 10.1.2

Replace $r$ with $- 4$ in the original inequality.

${(- 4)}^{2} > 3 (- 4) + 10$

Step 10.1.3

The left side $16$ is greater than the right side $- 2$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $- 2 < r < 5$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- 2 < r < 5$ and see if this value makes the original inequality true.

$r = 0$

Step 10.2.2

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

${(0)}^{2} > 3 (0) + 10$

Step 10.2.3

The left side $0$ is not greater than the right side $10$ , which means that the given statement is false.

False

Step 10.3

Test a value on the interval $r > 5$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $r > 5$ and see if this value makes the original inequality true.

$r = 8$

Step 10.3.2

Replace $r$ with $8$ in the original inequality.

${(8)}^{2} > 3 (8) + 10$

Step 10.3.3

The left side $64$ is greater than the right side $34$ , which means that the given statement is always true.

True

Step 10.4

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

$r < - 2$ True

$- 2 < r < 5$ False

$r > 5$ True

$r < - 2$ True

$- 2 < r < 5$ False

$r > 5$ True

Step 11

The solution consists of all of the true intervals.

$r < - 2$ or $r > 5$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$r < - 2 or r > 5$

Interval Notation:

$(- \infty, - 2) \cup (5, \infty)$

Step 13