Solve the Inequality for x -3x^2+25x-54>-8x

$- 3 x^{2} + 25 x - 54 > - 8 x$

Step 1

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

Add $8 x$ to both sides of the inequality.

$- 3 x^{2} + 25 x - 54 + 8 x > 0$

Add $25 x$ and $8 x$ .

$- 3 x^{2} + 33 x - 54 > 0$

Step 2

Convert the inequality to an equation.

$- 3 x^{2} + 33 x - 54 = 0$

Step 3

Factor the left side of the equation.

Factor $- 3$ out of $- 3 x^{2} + 33 x - 54$ .

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

$- 3 x^{2} + 33 x - 54 = 0$

Factor $- 3$ out of $33 x$ .

$- 3 x^{2} - 3 (- 11 x) - 54 = 0$

Factor $- 3$ out of $- 54$ .

$- 3 x^{2} - 3 (- 11 x) - 3 \cdot 18 = 0$

Factor $- 3$ out of $- 3 (x^{2}) - 3 (- 11 x)$ .

$- 3 (x^{2} - 11 x) - 3 \cdot 18 = 0$

Factor $- 3$ out of $- 3 (x^{2} - 11 x) - 3 (18)$ .

$- 3 (x^{2} - 11 x + 18) = 0$

Factor.

Factor $x^{2} - 11 x + 18$ 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 $- 11$ .

$- 9, - 2$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Step 4

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

$x - 9 = 0$

$x - 2 = 0$

Step 5

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 6

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 7

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

$x = 9, 2$

Step 8

Use each root to create test intervals.

$x < 2$

$2 < x < 9$

$x > 9$

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 $x < 2$ 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.

$x = 0$

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

$- 3 {(0)}^{2} + 25 (0) - 54 > - 8 \cdot 0$

The left side $- 54$ is not greater than the right side $0$ , 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.

$x = 6$

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

$- 3 {(6)}^{2} + 25 (6) - 54 > - 8 \cdot 6$

The left side $- 12$ is greater than the right side $- 48$ , 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.

$- 3 {(12)}^{2} + 25 (12) - 54 > - 8 \cdot 12$

The left side $- 186$ is not greater than the right side $- 96$ , 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 10

The solution consists of all of the true intervals.

$2 < x < 9$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$2 < x < 9$

Interval Notation:

$(2, 9)$

Step 12

Solve the Inequality for x -3x^2+25x-54>-8x

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