https://file.aiquickdraw.com/mathbot/file/1726206291836vq7d13n8.jpg

Let's work through the problem shown in the image step-by-step.

The inequality given is:

\[
x^2 + 18 > 11x
\]

First, let's rewrite the inequality to get everything on one side:

\[
x^2 - 11x + 18 > 0
\]

Now, solve the quadratic equation \(x^2 - 11x + 18 = 0\) by factoring:

\[
x^2 - 11x + 18 = (x - 9)(x - 2)
\]

So, the inequality becomes:

\[
(x - 9)(x - 2) > 0
\]

Now, we need to determine where the product of these two factors is positive. To do that, we can create a sign chart. The critical points, where the expression equals 0, are \(x = 9\) and \(x = 2\). These points divide the number line into three intervals: \((- \infty, 2)\), \((2, 9)\), and \((9, \infty)\).

Next, test a value in each interval:

1. For \(x < 2\) (e.g., \(x = 0\)):
   \[
   (0 - 9)(0 - 2) = (-9)(-2) = 18 > 0
   \]
   So the expression is positive in this interval.

2. For \(2 < x < 9\) (e.g., \(x = 5\)):
   \[
   (5 - 9)(5 - 2) = (-4)(3) = -12 < 0
   \]
   So the expression is negative in this interval.

3. For \(x > 9\) (e.g., \(x = 10\)):
   \[
   (10 - 9)(10 - 2) = (1)(8) = 8 > 0
   \]
   So the expression is positive in this interval.

Since we are interested in where the expression is positive, the solution is:

\[
x \in (-\infty, 2) \cup (9, \infty)
\]

Now, matching this with the options in the image, the correct answer is option **B**:

\[
x < 2 \text{ or } x > 9
\]

Do you have any questions about this solution, or would you like more details?

Here are 5 related questions for further practice:
1. How do you factor a quadratic equation like \(x^2 - 11x + 18 = 0\)?
2. What is a sign chart, and how is it used to solve inequalities?
3. What are the critical points of a quadratic inequality, and how are they found?
4. Why do we test intervals between critical points when solving inequalities?
5. How would the solution change if the inequality was \((x - 9)(x - 2) \geq 0\)?

**Tip:** Always move all terms to one side of the inequality before solving quadratic inequalities.

Solve the inequality: x^2 + 18 > 11x. Use a sign chart to express the solution in inequality and interval notation.

Learn how to solve the quadratic inequality x^2 + 18 > 11x using a sign chart. This solution includes factoring the quadratic expression, creating a sign chart, and determining the intervals where the inequality holds. Suitable for high school students in Grades 9-12.

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