Algebra Examples

$- x^{2} + x + 6 < 0$

Step 1

Convert the inequality to an equation.

$- x^{2} + x + 6 = 0$

Step 2

Factor the left side of the equation.

Tap for more steps...

Step 2.1

Factor $- 1$ out of $- x^{2} + x + 6$ .

Tap for more steps...

Step 2.1.1

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

$- (x^{2}) + x + 6 = 0$

Step 2.1.2

Factor $- 1$ out of $x$ .

$- (x^{2}) - 1 (- x) + 6 = 0$

Step 2.1.3

Rewrite $6$ as $- 1 (- 6)$ .

$- (x^{2}) - 1 (- x) - 1 \cdot - 6 = 0$

Step 2.1.4

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

$- (x^{2} - x) - 1 \cdot - 6 = 0$

Step 2.1.5

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 6)$ .

$- (x^{2} - x - 6) = 0$

Step 2.2

Factor.

Tap for more steps...

Step 2.2.1

Factor $x^{2} - x - 6$ using the AC method.

Tap for more steps...

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2.1.2

Write the factored form using these integers.

$- ((x - 3) (x + 2)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$- (x - 3) (x + 2) = 0$

Step 3

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

$x - 3 = 0$

$x + 2 = 0$

Step 4

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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

Tap for more steps...

Step 5.1

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

$x + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

The final solution is all the values that make $- (x - 3) (x + 2) = 0$ true.

$x = 3, - 2$

Step 7

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 3$

$x > 3$

Step 8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 8.1

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

Tap for more steps...

Step 8.1.1

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

$x = - 4$

Step 8.1.2

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

$- {(- 4)}^{2} - 4 + 6 < 0$

Step 8.1.3

The left side $- 14$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.2

Test a value on the interval $- 2 < x < 3$ to see if it makes the inequality true.

Tap for more steps...

Step 8.2.1

Choose a value on the interval $- 2 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

Step 8.2.2

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

$- {(0)}^{2} + 0 + 6 < 0$

Step 8.2.3

The left side $6$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Tap for more steps...

Step 8.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 8.3.2

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

$- {(6)}^{2} + 6 + 6 < 0$

Step 8.3.3

The left side $- 24$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.4

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

$x < - 2$ True

$- 2 < x < 3$ False

$x > 3$ True

$x < - 2$ True

$- 2 < x < 3$ False

$x > 3$ True

Step 9

The solution consists of all of the true intervals.

$x < - 2$ or $x > 3$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 3$

Interval Notation:

$(- \infty, - 2) \cup (3, \infty)$

Step 11