Algebra Examples

$- w^{2} + 14 w - 24 > 0$

Step 1

Convert the inequality to an equation.

$- w^{2} + 14 w - 24 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $- 1$ out of $- w^{2} + 14 w - 24$ .

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Step 2.1.1

Factor $- 1$ out of $- w^{2}$ .

$- (w^{2}) + 14 w - 24 = 0$

Step 2.1.2

Factor $- 1$ out of $14 w$ .

$- (w^{2}) - (- 14 w) - 24 = 0$

Step 2.1.3

Rewrite $- 24$ as $- 1 (24)$ .

$- (w^{2}) - (- 14 w) - 1 \cdot 24 = 0$

Step 2.1.4

Factor $- 1$ out of $- (w^{2}) - (- 14 w)$ .

$- (w^{2} - 14 w) - 1 \cdot 24 = 0$

Step 2.1.5

Factor $- 1$ out of $- (w^{2} - 14 w) - 1 (24)$ .

$- (w^{2} - 14 w + 24) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $w^{2} - 14 w + 24$ using the AC method.

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Step 2.2.1.1

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 $24$ and whose sum is $- 14$ .

$- 12, - 2$

Step 2.2.1.2

Write the factored form using these integers.

$- ((w - 12) (w - 2)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$- (w - 12) (w - 2) = 0$

Step 3

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

$w - 12 = 0$

$w - 2 = 0$

Step 4

Set $w - 12$ equal to $0$ and solve for $w$ .

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Step 4.1

Set $w - 12$ equal to $0$ .

$w - 12 = 0$

Step 4.2

Add $12$ to both sides of the equation.

$w = 12$

Step 5

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

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Step 5.1

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

$w - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$w = 2$

Step 6

The final solution is all the values that make $- (w - 12) (w - 2) = 0$ true.

$w = 12, 2$

Step 7

Use each root to create test intervals.

$w < 2$

$2 < w < 12$

$w > 12$

Step 8

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 8.1

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

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Step 8.1.1

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

$w = 0$

Step 8.1.2

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

$- {(0)}^{2} + 14 (0) - 24 > 0$

Step 8.1.3

The left side $- 24$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $2 < w < 12$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $2 < w < 12$ and see if this value makes the original inequality true.

$w = 7$

Step 8.2.2

Replace $w$ with $7$ in the original inequality.

$- {(7)}^{2} + 14 (7) - 24 > 0$

Step 8.2.3

The left side $25$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

Test a value on the interval $w > 12$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $w > 12$ and see if this value makes the original inequality true.

$w = 14$

Step 8.3.2

Replace $w$ with $14$ in the original inequality.

$- {(14)}^{2} + 14 (14) - 24 > 0$

Step 8.3.3

The left side $- 24$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$w < 2$ False

$2 < w < 12$ True

$w > 12$ False

$w < 2$ False

$2 < w < 12$ True

$w > 12$ False

Step 9

The solution consists of all of the true intervals.

$2 < w < 12$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$2 < w < 12$

Interval Notation:

$(2, 12)$

Step 11