Solve for x x^2+7x+6<0

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

Step 1

Convert the inequality to an equation.

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

Step 2

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.

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

Step 3

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

$x + 1 = 0$

$x + 6 = 0$

Step 4

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.

$x = - 1$

Step 5

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.

$x = - 6$

Step 6

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

$x = - 1, - 6$

Step 7

Use each root to create test intervals.

$x < - 6$

$- 6 < x < - 1$

$x > - 1$

Step 8

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 $x < - 6$ to see if it makes the inequality true.

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

$x = - 8$

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

${(- 8)}^{2} + 7 (- 8) + 6 < 0$

The left side $14$ is not less than the right side $0$ , which means that the given statement is false.

False

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.

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

The left side $- 6$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 0$

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

${(0)}^{2} + 7 (0) + 6 < 0$

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.

$x < - 6$ False

$- 6 < x < - 1$ True

$x > - 1$ False

$x < - 6$ False

$- 6 < x < - 1$ True

$x > - 1$ False

Step 9

The solution consists of all of the true intervals.

$- 6 < x < - 1$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 6 < x < - 1$

Interval Notation:

$(- 6, - 1)$

Step 11

Solve for x x^2+7x+6<0

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