Solve the Inequality for x 3x^2+38>24x-7

$3 x^{2} + 38 > 24 x - 7$

Step 1

Subtract $24 x$ from both sides of the inequality.

$3 x^{2} + 38 - 24 x > - 7$

Step 2

Convert the inequality to an equation.

$3 x^{2} + 38 - 24 x = - 7$

Step 3

Add $7$ to both sides of the equation.

$3 x^{2} + 38 - 24 x + 7 = 0$

Step 4

Add $38$ and $7$ .

$3 x^{2} - 24 x + 45 = 0$

Step 5

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} - 24 x + 45$ .

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

$3 (x^{2}) - 24 x + 45 = 0$

Factor $3$ out of $- 24 x$ .

$3 (x^{2}) + 3 (- 8 x) + 45 = 0$

Factor $3$ out of $45$ .

$3 x^{2} + 3 (- 8 x) + 3 \cdot 15 = 0$

Factor $3$ out of $3 x^{2} + 3 (- 8 x)$ .

$3 (x^{2} - 8 x) + 3 \cdot 15 = 0$

Factor $3$ out of $3 (x^{2} - 8 x) + 3 \cdot 15$ .

$3 (x^{2} - 8 x + 15) = 0$

Factor.

Factor $x^{2} - 8 x + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Step 6

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

$x - 5 = 0$

$x - 3 = 0$

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 $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 9

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

$x = 5, 3$

Step 10

Use each root to create test intervals.

$x < 3$

$3 < x < 5$

$x > 5$

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 $x < 3$ to see if it makes the inequality true.

Choose a value on the interval $x < 3$ and see if this value makes the original inequality true.

$x = 0$

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

$3 {(0)}^{2} + 38 > 24 (0) - 7$

The left side $38$ is greater than the right side $- 7$ , which means that the given statement is always true.

True

Test a value on the interval $3 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $3 < x < 5$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$3 {(4)}^{2} + 38 > 24 (4) - 7$

The left side $86$ is not greater than the right side $89$ , 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.

$3 {(8)}^{2} + 38 > 24 (8) - 7$

The left side $230$ is greater than the right side $185$ , which means that the given statement is always true.

True

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

$x < 3$ True

$3 < x < 5$ False

$x > 5$ True

$x < 3$ True

$3 < x < 5$ False

$x > 5$ True

Step 12

The solution consists of all of the true intervals.

$x < 3$ or $x > 5$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x < 3 or x > 5$

Interval Notation:

$(- \infty, 3) \cup (5, \infty)$

Step 14

Solve the Inequality for x 3x^2+38>24x-7

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