Algebra Examples

$- 2 x^{2} - 25 x < 5 x + 108$

Step 1

Move all terms containing $x$ to the left side of the inequality.

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Step 1.1

Subtract $5 x$ from both sides of the inequality.

$- 2 x^{2} - 25 x - 5 x < 108$

Step 1.2

Subtract $5 x$ from $- 25 x$ .

$- 2 x^{2} - 30 x < 108$

Step 2

Convert the inequality to an equation.

$- 2 x^{2} - 30 x = 108$

Step 3

Subtract $108$ from both sides of the equation.

$- 2 x^{2} - 30 x - 108 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 2$ out of $- 2 x^{2} - 30 x - 108$ .

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Step 4.1.1

Factor $- 2$ out of $- 2 x^{2}$ .

$- 2 x^{2} - 30 x - 108 = 0$

Step 4.1.2

Factor $- 2$ out of $- 30 x$ .

$- 2 x^{2} - 2 (15 x) - 108 = 0$

Step 4.1.3

Factor $- 2$ out of $- 108$ .

$- 2 x^{2} - 2 (15 x) - 2 \cdot 54 = 0$

Step 4.1.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (15 x)$ .

$- 2 (x^{2} + 15 x) - 2 \cdot 54 = 0$

Step 4.1.5

Factor $- 2$ out of $- 2 (x^{2} + 15 x) - 2 (54)$ .

$- 2 (x^{2} + 15 x + 54) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} + 15 x + 54$ using the AC method.

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Step 4.2.1.1

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 $54$ and whose sum is $15$ .

$6, 9$

Step 4.2.1.2

Write the factored form using these integers.

$- 2 ((x + 6) (x + 9)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- 2 (x + 6) (x + 9) = 0$

Step 5

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

$x + 6 = 0$

$x + 9 = 0$

Step 6

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 6.2

Subtract $6$ from both sides of the equation.

$x = - 6$

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 $- 2 (x + 6) (x + 9) = 0$ true.

$x = - 6, - 9$

Step 9

Use each root to create test intervals.

$x < - 9$

$- 9 < x < - 6$

$x > - 6$

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.

$- 2 {(- 12)}^{2} - 25 \cdot - 12 < 5 (- 12) + 108$

Step 10.1.3

The left side $12$ is less than the right side $48$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $- 9 < x < - 6$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- 9 < x < - 6$ and see if this value makes the original inequality true.

$x = - 8$

Step 10.2.2

Replace $x$ with $- 8$ in the original inequality.

$- 2 {(- 8)}^{2} - 25 \cdot - 8 < 5 (- 8) + 108$

Step 10.2.3

The left side $72$ is not less than the right side $68$ , which means that the given statement is false.

False

Step 10.3

Test a value on the interval $x > - 6$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $x > - 6$ and see if this value makes the original inequality true.

$x = 0$

Step 10.3.2

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

$- 2 {(0)}^{2} - 25 \cdot 0 < 5 (0) + 108$

Step 10.3.3

The left side $0$ is less than the right side $108$ , 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

$- 9 < x < - 6$ False

$x > - 6$ True

$x < - 9$ True

$- 9 < x < - 6$ False

$x > - 6$ True

Step 11

The solution consists of all of the true intervals.

$x < - 9$ or $x > - 6$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < - 9 or x > - 6$

Interval Notation:

$(- \infty, - 9) \cup (- 6, \infty)$

Step 13