Precalculus Examples

$x^{2} + 2 x \geq 8$

Step 1

Convert the inequality to an equation.

$x^{2} + 2 x = 8$

Step 2

Subtract $8$ from both sides of the equation.

$x^{2} + 2 x - 8 = 0$

Step 3

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 3.1

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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 3.2

Write the factored form using these integers.

$(x - 2) (x + 4) = 0$

Step 4

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

$x - 2 = 0$

$x + 4 = 0$

Step 5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

The final solution is all the values that make $(x - 2) (x + 4) = 0$ true.

$x = 2, - 4$

Step 8

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 2$

$x > 2$

Step 9

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 9.1

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 9.1.2

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

${(- 6)}^{2} + 2 (- 6) \geq 8$

Step 9.1.3

The left side $24$ is greater than the right side $8$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $- 4 < x < 2$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- 4 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 9.2.2

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

${(0)}^{2} + 2 (0) \geq 8$

Step 9.2.3

The left side $0$ is less than the right side $8$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 9.3.2

Replace $x$ with $4$ in the original inequality.

${(4)}^{2} + 2 (4) \geq 8$

Step 9.3.3

The left side $24$ is greater than the right side $8$ , which means that the given statement is always true.

True

Step 9.4

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

$x < - 4$ True

$- 4 < x < 2$ False

$x > 2$ True

$x < - 4$ True

$- 4 < x < 2$ False

$x > 2$ True

Step 10

The solution consists of all of the true intervals.

$x \leq - 4$ or $x \geq 2$

Step 11

Convert the inequality to interval notation.

$(- \infty, - 4] \cup [2, \infty)$

Step 12