Solve the Inequality for x -x^2+6x+57>7x+1

$- x^{2} + 6 x + 57 > 7 x + 1$

Step 1

Move all terms containing $x$ to the left side of the inequality.

Subtract $7 x$ from both sides of the inequality.

$- x^{2} + 6 x + 57 - 7 x > 1$

Subtract $7 x$ from $6 x$ .

$- x^{2} - x + 57 > 1$

Step 2

Convert the inequality to an equation.

$- x^{2} - x + 57 = 1$

Step 3

Subtract $1$ from both sides of the equation.

$- x^{2} - x + 57 - 1 = 0$

Step 4

Subtract $1$ from $57$ .

$- x^{2} - x + 56 = 0$

Step 5

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} - x + 56$ .

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

$- (x^{2}) - x + 56 = 0$

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

$- (x^{2}) - (x) + 56 = 0$

Rewrite $56$ as $- 1 (- 56)$ .

$- (x^{2}) - (x) - 1 \cdot - 56 = 0$

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

$- (x^{2} + x) - 1 \cdot - 56 = 0$

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 56)$ .

$- (x^{2} + x - 56) = 0$

Factor.

Factor $x^{2} + x - 56$ 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 $- 56$ and whose sum is $1$ .

$- 7, 8$

Write the factored form using these integers.

$- ((x - 7) (x + 8)) = 0$

Remove unnecessary parentheses.

$- (x - 7) (x + 8) = 0$

Step 6

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

$x - 7 = 0$

$x + 8 = 0$

Step 7

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 8

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.

$x = - 8$

Step 9

The final solution is all the values that make $- (x - 7) (x + 8) = 0$ true.

$x = 7, - 8$

Step 10

Use each root to create test intervals.

$x < - 8$

$- 8 < x < 7$

$x > 7$

Step 11

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.

$- {(- 10)}^{2} + 6 (- 10) + 57 > 7 (- 10) + 1$

The left side $- 103$ is not greater than the right side $- 69$ , which means that the given statement is false.

False

Test a value on the interval $- 8 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $- 8 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

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

$- {(0)}^{2} + 6 (0) + 57 > 7 (0) + 1$

The left side $57$ is greater than the right side $1$ , 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.

$- {(10)}^{2} + 6 (10) + 57 > 7 (10) + 1$

The left side $17$ is not greater than the right side $71$ , which means that the given statement is false.

False

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

$x < - 8$ False

$- 8 < x < 7$ True

$x > 7$ False

$x < - 8$ False

$- 8 < x < 7$ True

$x > 7$ False

Step 12

The solution consists of all of the true intervals.

$- 8 < x < 7$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 8 < x < 7$

Interval Notation:

$(- 8, 7)$

Step 14

Solve the Inequality for x -x^2+6x+57>7x+1

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