Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.
StartFraction x plus 6 Over x minus 3 EndFraction greater than 0


We are given the rational inequality:

\[
\frac{x + 6}{x - 3} > 0
\]

### Step 1: Determine where the expression is undefined
The expression is undefined where the denominator is zero. Set the denominator equal to zero and solve for \(x\):

\[
x - 3 = 0 \implies x = 3
\]

So, the expression is undefined at \(x = 3\).

### Step 2: Solve the inequality
To solve \(\frac{x + 6}{x - 3} > 0\), we need to find where the expression is positive.

#### Find the critical points
The numerator \(x + 6 = 0\) gives \(x = -6\).

Thus, the critical points are \(x = -6\) (where the numerator is zero) and \(x = 3\) (where the expression is undefined).

#### Divide the real line into intervals
The critical points divide the real number line into three intervals:

1. \( (-\infty, -6) \)
2. \( (-6, 3) \)
3. \( (3, \infty) \)

#### Test the sign of the expression in each interval
- **Interval 1: \( (-\infty, -6) \)**

  Choose \(x = -7\):

  \[
  \frac{-7 + 6}{-7 - 3} = \frac{-1}{-10} = \frac{1}{10} > 0
  \]

  The expression is positive in this interval.

- **Interval 2: \( (-6, 3) \)**

  Choose \(x = 0\):

  \[
  \frac{0 + 6}{0 - 3} = \frac{6}{-3} = -2 < 0
  \]

  The expression is negative in this interval.

- **Interval 3: \( (3, \infty) \)**

  Choose \(x = 4\):

  \[
  \frac{4 + 6}{4 - 3} = \frac{10}{1} = 10 > 0
  \]

  The expression is positive in this interval.

### Step 3: Analyze the inequality

We are looking for where the expression is **greater than zero**, so the solution will include the intervals where the expression is positive.

- In \( (-\infty, -6) \), the expression is positive.
- In \( (3, \infty) \), the expression is positive.

At \(x = -6\), the numerator is zero, so \(\frac{x + 6}{x - 3} = 0\). Since we are looking for where the expression is **greater** than zero (not greater than or equal to), \(x = -6\) is **not** included in the solution.

At \(x = 3\), the expression is undefined, so \(x = 3\) is also **not** included in the solution.

### Step 4: Write the solution in interval notation

The solution is:

\[
(-\infty, -6) \cup (3, \infty)
\]

### Step 5: Graph the solution on a real number line

- Draw open circles at \(x = -6\) and \(x = 3\) to indicate these points are not included.
- Shade the region to the left of \(x = -6\) and the region to the right of \(x = 3\).

---

Would you like me to generate the graph or explain any steps further?

Here are 5 related questions to explore further:
1. How would the solution change if the inequality were \(\frac{x + 6}{x - 3} \geq 0\)?
2. How do you handle rational inequalities with more complicated numerators and denominators?
3. What would happen if the inequality were \(\frac{x + 6}{x - 3} < 0\)?
4. How do we solve rational inequalities with quadratic expressions in the denominator?
5. How can we confirm our solution using a different method like a sign chart?

**Tip**: When solving rational inequalities, always identify where the expression is undefined first, as these values affect the solution set.

Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. StartFraction x plus 6 Over x minus 3 EndFraction greater than 0.

Learn how to solve the rational inequality (x + 6)/(x - 3) > 0 and express the solution set in interval notation. This step-by-step guide explains how to find critical points, test intervals, and graph the solution on a number line. Suitable for Grades 9-11 students studying rational inequalities.

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