To solve the inequality:

$\frac{x + 6}{x - 7} \leq 0$

Step 1: Identify the critical points

The numerator, $x + 6$, equals 0 when $x = -6$, and the denominator, $x - 7$, equals 0 when $x = 7$.

Thus, the critical points are $x = -6$ and $x = 7$. These points divide the real line into three intervals:

1. $(-∞, -6)$
2. $(-6, 7)$
3. $(7, ∞)$

Step 2: Determine the signs of the factors in each interval

We will analyze the signs of $x + 6$ and $x - 7$ in each interval.

Interval 1: $(-∞, -6)$

• $x + 6$ is negative (since for $x < -6$, $x + 6 < 0$).
• $x - 7$ is negative (since for $x < 7$, $x - 7 < 0$).
• The product of the two factors is positive, since negative divided by negative is positive.

Interval 2: $(-6, 7)$

• $x + 6$ is positive (since for $x > -6$, $x + 6 > 0$).
• $x - 7$ is negative (since for $x < 7$, $x - 7 < 0$).
• The product of the two factors is negative, since positive divided by negative is negative.

Interval 3: $(7, ∞)$

• $x + 6$ is positive (since for $x > -6$, $x + 6 > 0$).
• $x - 7$ is positive (since for $x > 7$, $x - 7 > 0$).
• The product of the two factors is positive, since positive divided by positive is positive.

Step 3: Fill out the table

Here’s the table with the signs of $x + 6$, $x - 7$, and their quotient:

Interval	$-∞, -6$	$-6, 7$	$7, ∞$
Sign of $x + 6$	Negative	Positive	Positive
Sign of $x - 7$	Negative	Negative	Positive
Sign of $\frac{x + 6}{x - 7}$	Positive	Negative	Positive

Step 4: Analyze the endpoints

• At $x = -6$, $x + 6 = 0$ and $x - 7 = -13$. Thus, $\frac{x + 6}{x - 7} = 0$, which satisfies the inequality $\frac{x + 6}{x - 7} \leq 0$.
• At $x = 7$, $x + 6 = 13$ and $x - 7 = 0$. The expression is undefined because we cannot divide by zero.

Therefore, $x = 7$ does not satisfy the inequality, while $x = -6$ does.

Step 5: Find the solution

• The inequality holds where the expression is less than or equal to zero. From the table, this happens in the interval $(-6, 7)$, including $x = -6$ but excluding $x = 7$ (because $x = 7$ is undefined).

Thus, the solution in interval notation is:

$\boxed{[-6, 7)}$

Summary

• Endpoints that fail to satisfy the inequality: $x = 7$.
• Solution in interval notation: $[-6, 7)$.

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Would you like more details on how to approach this kind of inequality? Here are a few related questions to help you practice further:

1. How do we handle inequalities with quadratic expressions in the numerator and denominator?
2. What is the difference between a strict inequality and a non-strict inequality in rational expressions?
3. How would the solution change if the inequality were $\frac{x + 6}{x - 7} > 0$?
4. How do we find the sign of a rational expression without explicitly testing each interval?
5. How do the behaviors of rational expressions near their undefined points influence the solution?

Tip: When solving rational inequalities, always check the endpoints where the expression is undefined (such as division by zero) and verify if they are part of the solution set.