Solve the Inequality for x -3x^2+28x>=x+54

$- 3 x^{2} + 28 x \geq x + 54$

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

Subtract $x$ from both sides of the inequality.

$- 3 x^{2} + 28 x - x \geq 54$

Subtract $x$ from $28 x$ .

$- 3 x^{2} + 27 x \geq 54$

Convert the inequality to an equation.

$- 3 x^{2} + 27 x = 54$

Subtract $54$ from both sides of the equation.

$- 3 x^{2} + 27 x - 54 = 0$

Factor the left side of the equation.

Factor $- 3$ out of $- 3 x^{2} + 27 x - 54$ .

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

$- 3 x^{2} + 27 x - 54 = 0$

Factor $- 3$ out of $27 x$ .

$- 3 x^{2} - 3 (- 9 x) - 54 = 0$

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

$- 3 x^{2} - 3 (- 9 x) - 3 \cdot 18 = 0$

Factor $- 3$ out of $- 3 (x^{2}) - 3 (- 9 x)$ .

$- 3 (x^{2} - 9 x) - 3 \cdot 18 = 0$

Factor $- 3$ out of $- 3 (x^{2} - 9 x) - 3 (18)$ .

$- 3 (x^{2} - 9 x + 18) = 0$

Factor.

Factor $x^{2} - 9 x + 18$ 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 $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$- 3 ((x - 6) (x - 3)) = 0$

Remove unnecessary parentheses.

$- 3 (x - 6) (x - 3) = 0$

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

$x - 6 = 0$

$x - 3 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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.

$x = 3$

The final solution is all the values that make $- 3 (x - 6) (x - 3) = 0$ true.

$x = 6, 3$

Use each root to create test intervals.

$x < 3$

$3 < x < 6$

$x > 6$

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.

$- 3 {(0)}^{2} + 28 (0) \geq (0) + 54$

The left side $0$ is less than the right side $54$ , which means that the given statement is false.

False

Test a value on the interval $3 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $3 < x < 6$ and see if this value makes the original inequality true.

$x = 4$

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

$- 3 {(4)}^{2} + 28 (4) \geq (4) + 54$

The left side $64$ is greater than the right side $58$ , which means that the given statement is always true.

True

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

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

$- 3 {(8)}^{2} + 28 (8) \geq (8) + 54$

The left side $32$ is less than the right side $62$ , which means that the given statement is false.

False

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

$x < 3$ False

$3 < x < 6$ True

$x > 6$ False

$x < 3$ False

$3 < x < 6$ True

$x > 6$ False

The solution consists of all of the true intervals.

$3 \leq x \leq 6$

The result can be shown in multiple forms.

Inequality Form:

$3 \leq x \leq 6$

Interval Notation:

$[3, 6]$

Solve the Inequality for x -3x^2+28x>=x+54

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