Solve the Inequality for x -3x^2+84<=-9x

$- 3 x^{2} + 84 \leq - 9 x$

Step 1

Add $9 x$ to both sides of the inequality.

$- 3 x^{2} + 84 + 9 x \leq 0$

Step 2

Convert the inequality to an equation.

$- 3 x^{2} + 84 + 9 x = 0$

Step 3

Factor the left side of the equation.

Factor $- 3$ out of $- 3 x^{2} + 84 + 9 x$ .

Move $84$ .

$- 3 x^{2} + 9 x + 84 = 0$

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

$- 3 x^{2} + 9 x + 84 = 0$

Factor $- 3$ out of $9 x$ .

$- 3 x^{2} - 3 (- 3 x) + 84 = 0$

Factor $- 3$ out of $84$ .

$- 3 x^{2} - 3 (- 3 x) - 3 \cdot - 28 = 0$

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

$- 3 (x^{2} - 3 x) - 3 \cdot - 28 = 0$

Factor $- 3$ out of $- 3 (x^{2} - 3 x) - 3 (- 28)$ .

$- 3 (x^{2} - 3 x - 28) = 0$

Factor.

Factor $x^{2} - 3 x - 28$ 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 $- 28$ and whose sum is $- 3$ .

$- 7, 4$

Write the factored form using these integers.

$- 3 ((x - 7) (x + 4)) = 0$

Remove unnecessary parentheses.

$- 3 (x - 7) (x + 4) = 0$

Step 4

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

$x - 7 = 0$

$x + 4 = 0$

Step 5

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 6

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

The final solution is all the values that make $- 3 (x - 7) (x + 4) = 0$ true.

$x = 7, - 4$

Step 8

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 7$

$x > 7$

Step 9

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 $x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

$- 3 {(- 6)}^{2} + 84 \leq - 9 \cdot - 6$

The left side $- 24$ is less than the right side $54$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

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

$- 3 {(0)}^{2} + 84 \leq - 9 \cdot 0$

The left side $84$ is greater than the right side $0$ , which means that the given statement is false.

False

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.

$- 3 {(10)}^{2} + 84 \leq - 9 \cdot 10$

The left side $- 216$ is less than the right side $- 90$ , which means that the given statement is always true.

True

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

$x < - 4$ True

$- 4 < x < 7$ False

$x > 7$ True

$x < - 4$ True

$- 4 < x < 7$ False

$x > 7$ True

Step 10

The solution consists of all of the true intervals.

$x \leq - 4$ or $x \geq 7$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 4 or x \geq 7$

Interval Notation:

$(- \infty, - 4] \cup [7, \infty)$

Step 12

Solve the Inequality for x -3x^2+84<=-9x

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