Solve the Inequality for x -3x^2+39x-126>-6

$- 3 x^{2} + 39 x - 126 > - 6$

Step 1

Convert the inequality to an equation.

$- 3 x^{2} + 39 x - 126 = - 6$

Step 2

Add $6$ to both sides of the equation.

$- 3 x^{2} + 39 x - 126 + 6 = 0$

Step 3

Add $- 126$ and $6$ .

$- 3 x^{2} + 39 x - 120 = 0$

Step 4

Factor the left side of the equation.

Factor $- 3$ out of $- 3 x^{2} + 39 x - 120$ .

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

$- 3 x^{2} + 39 x - 120 = 0$

Factor $- 3$ out of $39 x$ .

$- 3 x^{2} - 3 (- 13 x) - 120 = 0$

Factor $- 3$ out of $- 120$ .

$- 3 x^{2} - 3 (- 13 x) - 3 \cdot 40 = 0$

Factor $- 3$ out of $- 3 (x^{2}) - 3 (- 13 x)$ .

$- 3 (x^{2} - 13 x) - 3 \cdot 40 = 0$

Factor $- 3$ out of $- 3 (x^{2} - 13 x) - 3 (40)$ .

$- 3 (x^{2} - 13 x + 40) = 0$

Factor.

Factor $x^{2} - 13 x + 40$ 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 $40$ and whose sum is $- 13$ .

$- 8, - 5$

Write the factored form using these integers.

$- 3 ((x - 8) (x - 5)) = 0$

Remove unnecessary parentheses.

$- 3 (x - 8) (x - 5) = 0$

Step 5

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

$x - 8 = 0$

$x - 5 = 0$

Step 6

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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

The final solution is all the values that make $- 3 (x - 8) (x - 5) = 0$ true.

$x = 8, 5$

Step 9

Use each root to create test intervals.

$x < 5$

$5 < x < 8$

$x > 8$

Step 10

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 $x < 5$ 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.

$x = 0$

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

$- 3 {(0)}^{2} + 39 (0) - 126 > - 6$

The left side $- 126$ is not greater than the right side $- 6$ , which means that the given statement is false.

False

Test a value on the interval $5 < x < 8$ to see if it makes the inequality true.

Choose a value on the interval $5 < x < 8$ and see if this value makes the original inequality true.

$x = 6$

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

$- 3 {(6)}^{2} + 39 (6) - 126 > - 6$

The left side $0$ is greater than the right side $- 6$ , which means that the given statement is always true.

True

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.

$- 3 {(10)}^{2} + 39 (10) - 126 > - 6$

The left side $- 36$ is not greater than the right side $- 6$ , which means that the given statement is false.

False

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

$x < 5$ False

$5 < x < 8$ True

$x > 8$ False

$x < 5$ False

$5 < x < 8$ True

$x > 8$ False

Step 11

The solution consists of all of the true intervals.

$5 < x < 8$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$5 < x < 8$

Interval Notation:

$(5, 8)$

Step 13

Solve the Inequality for x -3x^2+39x-126>-6

Related Questions

Support LCMGCF.com by disabling your adblocker.