Solve the Inequality for x -3x^2+16x+53>-5x-1

$- 3 x^{2} + 16 x + 53 > - 5 x - 1$

Step 1

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

Add $5 x$ to both sides of the inequality.

$- 3 x^{2} + 16 x + 53 + 5 x > - 1$

Add $16 x$ and $5 x$ .

$- 3 x^{2} + 21 x + 53 > - 1$

Step 2

Convert the inequality to an equation.

$- 3 x^{2} + 21 x + 53 = - 1$

Step 3

Add $1$ to both sides of the equation.

$- 3 x^{2} + 21 x + 53 + 1 = 0$

Step 4

Add $53$ and $1$ .

$- 3 x^{2} + 21 x + 54 = 0$

Step 5

Factor the left side of the equation.

Factor $- 3$ out of $- 3 x^{2} + 21 x + 54$ .

Factor $- 3$ out of $- 3 x^{2}$ .

$- 3 x^{2} + 21 x + 54 = 0$

Factor $- 3$ out of $21 x$ .

$- 3 x^{2} - 3 (- 7 x) + 54 = 0$

Factor $- 3$ out of $54$ .

$- 3 x^{2} - 3 (- 7 x) - 3 \cdot - 18 = 0$

Factor $- 3$ out of $- 3 (x^{2}) - 3 (- 7 x)$ .

$- 3 (x^{2} - 7 x) - 3 \cdot - 18 = 0$

Factor $- 3$ out of $- 3 (x^{2} - 7 x) - 3 (- 18)$ .

$- 3 (x^{2} - 7 x - 18) = 0$

Factor.

Factor $x^{2} - 7 x - 18$ 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 $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Write the factored form using these integers.

$- 3 ((x - 9) (x + 2)) = 0$

Remove unnecessary parentheses.

$- 3 (x - 9) (x + 2) = 0$

Step 6

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

$x - 9 = 0$

$x + 2 = 0$

Step 7

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Step 8

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.

$x = - 2$

Step 9

The final solution is all the values that make $- 3 (x - 9) (x + 2) = 0$ true.

$x = 9, - 2$

Step 10

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 9$

$x > 9$

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 $x < - 2$ 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.

$- 3 {(- 4)}^{2} + 16 (- 4) + 53 > - 5 \cdot - 4 - 1$

The left side $- 59$ is not greater than the right side $19$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 9$ and see if this value makes the original inequality true.

$x = 0$

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

$- 3 {(0)}^{2} + 16 (0) + 53 > - 5 \cdot 0 - 1$

The left side $53$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Test a value on the interval $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$- 3 {(12)}^{2} + 16 (12) + 53 > - 5 \cdot 12 - 1$

The left side $- 187$ is not greater than the right side $- 61$ , which means that the given statement is false.

False

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

$x < - 2$ False

$- 2 < x < 9$ True

$x > 9$ False

$x < - 2$ False

$- 2 < x < 9$ True

$x > 9$ False

Step 12

The solution consists of all of the true intervals.

$- 2 < x < 9$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 9$

Interval Notation:

$(- 2, 9)$

Step 14

Solve the Inequality for x -3x^2+16x+53>-5x-1

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