Solve the Inequality for x x^2+5x+6<0

$x^{2} + 5 x + 6 < 0$

Convert the inequality to an equation.

$x^{2} + 5 x + 6 = 0$

Factor $x^{2} + 5 x + 6$ using the AC method.

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 $5$ .

$2, 3$

Write the factored form using these integers.

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

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

$x + 2 = 0$

$x + 3 = 0$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

$x = - 2, - 3$

Use each root to create test intervals.

$x < - 3$

$- 3 < x < - 2$

$x > - 2$

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

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

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

$x = - 6$

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

${(- 6)}^{2} + 5 (- 6) + 6 < 0$

The left side $12$ is not less than the right side $0$ , which means that the given statement is false.

False

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

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

$x = - 2.5$

Replace $x$ with $- 2.5$ in the original inequality.

${(- 2.5)}^{2} + 5 (- 2.5) + 6 < 0$

The left side $- 0.25$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 0$

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

${(0)}^{2} + 5 (0) + 6 < 0$

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

False

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

$x < - 3$ False

$- 3 < x < - 2$ True

$x > - 2$ False

$x < - 3$ False

$- 3 < x < - 2$ True

$x > - 2$ False

The solution consists of all of the true intervals.

$- 3 < x < - 2$

The result can be shown in multiple forms.

Inequality Form:

$- 3 < x < - 2$

Interval Notation:

$(- 3, - 2)$

Solve the Inequality for x x^2+5x+6<0

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