Solve the Inequality for x 3x^2-54x<=-3x-210

$3 x^{2} - 54 x \leq - 3 x - 210$

Step 1

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

Add $3 x$ to both sides of the inequality.

$3 x^{2} - 54 x + 3 x \leq - 210$

Add $- 54 x$ and $3 x$ .

$3 x^{2} - 51 x \leq - 210$

Step 2

Convert the inequality to an equation.

$3 x^{2} - 51 x = - 210$

Step 3

Add $210$ to both sides of the equation.

$3 x^{2} - 51 x + 210 = 0$

Step 4

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} - 51 x + 210$ .

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

$3 (x^{2}) - 51 x + 210 = 0$

Factor $3$ out of $- 51 x$ .

$3 (x^{2}) + 3 (- 17 x) + 210 = 0$

Factor $3$ out of $210$ .

$3 x^{2} + 3 (- 17 x) + 3 \cdot 70 = 0$

Factor $3$ out of $3 x^{2} + 3 (- 17 x)$ .

$3 (x^{2} - 17 x) + 3 \cdot 70 = 0$

Factor $3$ out of $3 (x^{2} - 17 x) + 3 \cdot 70$ .

$3 (x^{2} - 17 x + 70) = 0$

Factor.

Factor $x^{2} - 17 x + 70$ 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 $70$ and whose sum is $- 17$ .

$- 10, - 7$

Write the factored form using these integers.

$3 ((x - 10) (x - 7)) = 0$

Remove unnecessary parentheses.

$3 (x - 10) (x - 7) = 0$

Step 5

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

$x - 10 = 0$

$x - 7 = 0$

Step 6

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

Step 7

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 8

The final solution is all the values that make $3 (x - 10) (x - 7) = 0$ true.

$x = 10, 7$

Step 9

Use each root to create test intervals.

$x < 7$

$7 < x < 10$

$x > 10$

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 $x < 7$ 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.

$x = 0$

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

$3 {(0)}^{2} - 54 \cdot 0 \leq - 3 \cdot 0 - 210$

The left side $0$ is greater than the right side $- 210$ , which means that the given statement is false.

False

Test a value on the interval $7 < x < 10$ to see if it makes the inequality true.

Choose a value on the interval $7 < x < 10$ and see if this value makes the original inequality true.

$x = 8$

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

$3 {(8)}^{2} - 54 \cdot 8 \leq - 3 \cdot 8 - 210$

The left side $- 240$ is less than the right side $- 234$ , which means that the given statement is always true.

True

Test a value on the interval $x > 10$ to see if it makes the inequality true.

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$3 {(12)}^{2} - 54 \cdot 12 \leq - 3 \cdot 12 - 210$

The left side $- 216$ is greater than the right side $- 246$ , which means that the given statement is false.

False

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

$x < 7$ False

$7 < x < 10$ True

$x > 10$ False

$x < 7$ False

$7 < x < 10$ True

$x > 10$ False

Step 11

The solution consists of all of the true intervals.

$7 \leq x \leq 10$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$7 \leq x \leq 10$

Interval Notation:

$[7, 10]$

Step 13

Solve the Inequality for x 3x^2-54x<=-3x-210

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