Solve the Inequality for x -x^2+6x+7>=0

$- x^{2} + 6 x + 7 \geq 0$

Step 1

Convert the inequality to an equation.

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

Step 2

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} + 6 x + 7$ .

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

$- (x^{2}) + 6 x + 7 = 0$

Factor $- 1$ out of $6 x$ .

$- (x^{2}) - (- 6 x) + 7 = 0$

Rewrite $7$ as $- 1 (- 7)$ .

$- (x^{2}) - (- 6 x) - 1 \cdot - 7 = 0$

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

$- (x^{2} - 6 x) - 1 \cdot - 7 = 0$

Factor $- 1$ out of $- (x^{2} - 6 x) - 1 (- 7)$ .

$- (x^{2} - 6 x - 7) = 0$

Factor.

Factor $x^{2} - 6 x - 7$ 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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Write the factored form using these integers.

$- ((x - 7) (x + 1)) = 0$

Remove unnecessary parentheses.

$- (x - 7) (x + 1) = 0$

Step 3

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

$x - 7 = 0$

$x + 1 = 0$

Step 4

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 5

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 6

The final solution is all the values that make $- (x - 7) (x + 1) = 0$ true.

$x = 7, - 1$

Step 7

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 7$

$x > 7$

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 $x < - 1$ 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.

$x = - 4$

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

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

The left side $- 33$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

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

$- {(0)}^{2} + 6 (0) + 7 \geq 0$

The left side $7$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 7$ to see if it makes the inequality true.

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

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

$- {(10)}^{2} + 6 (10) + 7 \geq 0$

The left side $- 33$ is 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.

$x < - 1$ False

$- 1 < x < 7$ True

$x > 7$ False

$x < - 1$ False

$- 1 < x < 7$ True

$x > 7$ False

Step 9

The solution consists of all of the true intervals.

$- 1 \leq x \leq 7$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 1 \leq x \leq 7$

Interval Notation:

$[- 1, 7]$

Step 11

Solve the Inequality for x -x^2+6x+7>=0

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