Solve the Inequality for x -3x^2+39<=-9x+9

$- 3 x^{2} + 39 \leq - 9 x + 9$

Step 1

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

$- 3 x^{2} + 39 + 9 x \leq 9$

Step 2

Convert the inequality to an equation.

$- 3 x^{2} + 39 + 9 x = 9$

Step 3

Subtract $9$ from both sides of the equation.

$- 3 x^{2} + 39 + 9 x - 9 = 0$

Step 4

Subtract $9$ from $39$ .

$- 3 x^{2} + 9 x + 30 = 0$

Step 5

Factor the left side of the equation.

Factor $- 3$ out of $- 3 x^{2} + 9 x + 30$ .

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

$- 3 x^{2} + 9 x + 30 = 0$

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

$- 3 x^{2} - 3 (- 3 x) + 30 = 0$

Factor $- 3$ out of $30$ .

$- 3 x^{2} - 3 (- 3 x) - 3 \cdot - 10 = 0$

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

$- 3 (x^{2} - 3 x) - 3 \cdot - 10 = 0$

Factor $- 3$ out of $- 3 (x^{2} - 3 x) - 3 (- 10)$ .

$- 3 (x^{2} - 3 x - 10) = 0$

Factor.

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$- 3 ((x - 5) (x + 2)) = 0$

Remove unnecessary parentheses.

$- 3 (x - 5) (x + 2) = 0$

Step 6

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

$x - 5 = 0$

$x + 2 = 0$

Step 7

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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 $- 3 (x - 5) (x + 2) = 0$ true.

$x = 5, - 2$

Step 10

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 5$

$x > 5$

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.

$- 3 {(- 4)}^{2} + 39 \leq - 9 \cdot - 4 + 9$

The left side $- 9$ is less than the right side $45$ , which means that the given statement is always true.

True

Test a value on the interval $- 2 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 5$ and see if this value makes the original inequality true.

$x = 0$

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

$- 3 {(0)}^{2} + 39 \leq - 9 \cdot 0 + 9$

The left side $39$ is greater than the right side $9$ , which means that the given statement is false.

False

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

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

$- 3 {(8)}^{2} + 39 \leq - 9 \cdot 8 + 9$

The left side $- 153$ is less than the right side $- 63$ , which means that the given statement is always true.

True

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

$x < - 2$ True

$- 2 < x < 5$ False

$x > 5$ True

$x < - 2$ True

$- 2 < x < 5$ False

$x > 5$ True

Step 12

The solution consists of all of the true intervals.

$x \leq - 2$ or $x \geq 5$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 2 or x \geq 5$

Interval Notation:

$(- \infty, - 2] \cup [5, \infty)$

Step 14

Solve the Inequality for x -3x^2+39<=-9x+9

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