Let's analyze the two problems in the image.

1. Interval Notation from the Number Line:

The first part shows a number line with an open circle at $3$ and an arrow extending to the right, covering all numbers greater than $3$.

This implies that the solution set does not include $3$ itself but includes all values greater than $3$. Hence, in interval notation: $(3, \infty)$

2. Rational Inequality:

The inequality given is: $\frac{1}{x+3} > \frac{1}{x-4}$ To solve this inequality, follow these steps:

1. Rewrite the inequality: Subtract $\frac{1}{x-4}$ from both sides: $\frac{1}{x+3} - \frac{1}{x-4} > 0$

2. Find a common denominator: The common denominator is $(x+3)(x-4)$. So, $\frac{(x-4) - (x+3)}{(x+3)(x-4)} > 0$ Simplifying the numerator: $\frac{x-4 - x - 3}{(x+3)(x-4)} = \frac{-7}{(x+3)(x-4)} > 0$

3. Analyze the sign: The inequality $\frac{-7}{(x+3)(x-4)} > 0$ holds when the denominator $(x+3)(x-4)$ is negative (because $-7$ is negative).

   Therefore, we need to determine when $(x+3)(x-4) < 0$. The key points are $x = -3$ and $x = 4$. So, examine the sign of $(x+3)(x-4)$ in the intervals:

   • $x < -3$
   • $-3 < x < 4$
   • $x > 4$

   After checking the signs, the solution occurs when $x \in (-3, 4)$.

Thus, the solution in interval notation is: $(-3, 4)$ Now, selecting the correct graph, graph B matches the interval $(-3, 4)$, so the correct answer is B.

Would you like more detailed steps for solving rational inequalities?

Relative Questions:

1. How do we graph inequalities on a number line?
2. How can we identify when to use open or closed intervals in interval notation?
3. What are the steps for solving a rational inequality?
4. How do we find the common denominator in rational inequalities?
5. What is the importance of identifying critical points in inequality problems?

Tip:

When solving inequalities, always check the sign changes in the intervals around critical points to determine where the inequality holds.