Algebra Examples

$- x^{2} + 50 > - 3 x + 10$

Step 1

Add $3 x$ to both sides of the inequality.

$- x^{2} + 50 + 3 x > 10$

Step 2

Convert the inequality to an equation.

$- x^{2} + 50 + 3 x = 10$

Step 3

Subtract $10$ from both sides of the equation.

$- x^{2} + 50 + 3 x - 10 = 0$

Step 4

Subtract $10$ from $50$ .

$- x^{2} + 3 x + 40 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $- 1$ out of $- x^{2} + 3 x + 40$ .

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Step 5.1.1

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

$- (x^{2}) + 3 x + 40 = 0$

Step 5.1.2

Factor $- 1$ out of $3 x$ .

$- (x^{2}) - (- 3 x) + 40 = 0$

Step 5.1.3

Rewrite $40$ as $- 1 (- 40)$ .

$- (x^{2}) - (- 3 x) - 1 \cdot - 40 = 0$

Step 5.1.4

Factor $- 1$ out of $- (x^{2}) - (- 3 x)$ .

$- (x^{2} - 3 x) - 1 \cdot - 40 = 0$

Step 5.1.5

Factor $- 1$ out of $- (x^{2} - 3 x) - 1 (- 40)$ .

$- (x^{2} - 3 x - 40) = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor $x^{2} - 3 x - 40$ using the AC method.

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Step 5.2.1.1

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 $- 40$ and whose sum is $- 3$ .

$- 8, 5$

Step 5.2.1.2

Write the factored form using these integers.

$- ((x - 8) (x + 5)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$- (x - 8) (x + 5) = 0$

Step 6

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

$x - 8 = 0$

$x + 5 = 0$

Step 7

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 7.2

Add $8$ to both sides of the equation.

$x = 8$

Step 8

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 8.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 9

The final solution is all the values that make $- (x - 8) (x + 5) = 0$ true.

$x = 8, - 5$

Step 10

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 8$

$x > 8$

Step 11

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 11.1

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

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Step 11.1.1

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

$x = - 8$

Step 11.1.2

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

$- {(- 8)}^{2} + 50 > - 3 \cdot - 8 + 10$

Step 11.1.3

The left side $- 14$ is not greater than the right side $34$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $- 5 < x < 8$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $- 5 < x < 8$ and see if this value makes the original inequality true.

$x = 0$

Step 11.2.2

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

$- {(0)}^{2} + 50 > - 3 \cdot 0 + 10$

Step 11.2.3

The left side $50$ is greater than the right side $10$ , which means that the given statement is always true.

True

Step 11.3

Test a value on the interval $x > 8$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

Step 11.3.2

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

$- {(10)}^{2} + 50 > - 3 \cdot 10 + 10$

Step 11.3.3

The left side $- 50$ is not greater than the right side $- 20$ , which means that the given statement is false.

False

Step 11.4

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

$x < - 5$ False

$- 5 < x < 8$ True

$x > 8$ False

$x < - 5$ False

$- 5 < x < 8$ True

$x > 8$ False

Step 12

The solution consists of all of the true intervals.

$- 5 < x < 8$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x < 8$

Interval Notation:

$(- 5, 8)$

Step 14