Solve the Inequality for x (x^2+6x-16)/(x^2+7x+6)<0

$\frac{x^{2} + 6 x - 16}{x^{2} + 7 x + 6} < 0$

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} + 6 x - 16 = 0$

$x^{2} + 7 x + 6 = 0$

Factor $x^{2} + 6 x - 16$ 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 $- 16$ and whose sum is $6$ .

$- 2, 8$

Write the factored form using these integers.

$(x - 2) (x + 8) = 0$

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

$x - 2 = 0$

$x + 8 = 0$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $(x - 2) (x + 8) = 0$ true.

$x = 2, - 8$

Factor $x^{2} + 7 x + 6$ 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 $7$ .

$1, 6$

Write the factored form using these integers.

$(x + 1) (x + 6) = 0$

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

$x + 1 = 0$

$x + 6 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x + 1) (x + 6) = 0$ true.

$x = - 1, - 6$

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 2, - 8$

$x = - 1, - 6$

Consolidate the solutions.

$x = 2, - 8, - 1, - 6$

Find the domain of $\frac{x^{2} + 6 x - 16}{x^{2} + 7 x + 6}$ .

Set the denominator in $\frac{x^{2} + 6 x - 16}{x^{2} + 7 x + 6}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 7 x + 6 = 0$

Solve for $x$ .

Factor $x^{2} + 7 x + 6$ 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 $7$ .

$1, 6$

Write the factored form using these integers.

$(x + 1) (x + 6) = 0$

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

$x + 1 = 0$

$x + 6 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x + 1) (x + 6) = 0$ true.

$x = - 1, - 6$

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 6) \cup (- 6, - 1) \cup (- 1, \infty)$

Use each root to create test intervals.

$x < - 8$

$- 8 < x < - 6$

$- 6 < x < - 1$

$- 1 < x < 2$

$x > 2$

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 $x < - 8$ to see if it makes the inequality true.

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

$x = - 10$

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

$\frac{{(- 10)}^{2} + 6 (- 10) - 16}{{(- 10)}^{2} + 7 (- 10) + 6} < 0$

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

False

Test a value on the interval $- 8 < x < - 6$ to see if it makes the inequality true.

Choose a value on the interval $- 8 < x < - 6$ and see if this value makes the original inequality true.

$x = - 7$

Replace $x$ with $- 7$ in the original inequality.

$\frac{{(- 7)}^{2} + 6 (- 7) - 16}{{(- 7)}^{2} + 7 (- 7) + 6} < 0$

The left side $- 1.5$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 6 < x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

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

$\frac{{(- 4)}^{2} + 6 (- 4) - 16}{{(- 4)}^{2} + 7 (- 4) + 6} < 0$

The left side $4$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{{(0)}^{2} + 6 (0) - 16}{{(0)}^{2} + 7 (0) + 6} < 0$

The left side $- 2.\overline{6}$ 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.

$x = 4$

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

$\frac{{(4)}^{2} + 6 (4) - 16}{{(4)}^{2} + 7 (4) + 6} < 0$

The left side $0.48$ 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.

$x < - 8$ False

$- 8 < x < - 6$ True

$- 6 < x < - 1$ False

$- 1 < x < 2$ True

$x > 2$ False

$x < - 8$ False

$- 8 < x < - 6$ True

$- 6 < x < - 1$ False

$- 1 < x < 2$ True

$x > 2$ False

The solution consists of all of the true intervals.

$- 8 < x < - 6$ or $- 1 < x < 2$

The result can be shown in multiple forms.

Inequality Form:

$- 8 < x < - 6 or - 1 < x < 2$

Interval Notation:

$(- 8, - 6) \cup (- 1, 2)$

Solve the Inequality for x (x^2+6x-16)/(x^2+7x+6)<0

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