Let's solve the two inequalities one by one.

1) $\frac{x - 7}{x - 1} < 0$

This inequality asks where the expression is negative. Here's the step-by-step solution:

Step 1: Determine the critical points

The critical points are values of $x$ where the numerator or denominator is zero:

• Numerator: $x - 7 = 0 \implies x = 7$
• Denominator: $x - 1 = 0 \implies x = 1$

So, the critical points are $x = 7$ and $x = 1$. These points divide the real number line into three intervals:

• $(-\infty, 1)$
• $(1, 7)$
• $(7, \infty)$

Step 2: Test intervals

Now, test points from each interval to determine the sign of the expression $\frac{x - 7}{x - 1}$ in each region.

• For $x \in (-\infty, 1)$, choose $x = 0$: $\frac{0 - 7}{0 - 1} = \frac{-7}{-1} = 7 > 0$ (positive)

• For $x \in (1, 7)$, choose $x = 2$: $\frac{2 - 7}{2 - 1} = \frac{-5}{1} = -5 < 0$ (negative)

• For $x \in (7, \infty)$, choose $x = 8$: $\frac{8 - 7}{8 - 1} = \frac{1}{7} > 0$ (positive)

Step 3: Conclusion

The expression is negative in the interval $(1, 7)$, so the solution is: $1 < x < 7$ Since the inequality is strict ($< 0$), the solution does not include the points $x = 1$ and $x = 7$.

2) $\frac{x + 5}{x - 4} \leq 0$

This inequality asks where the expression is non-positive (i.e., less than or equal to zero).

Step 1: Determine the critical points

• Numerator: $x + 5 = 0 \implies x = -5$
• Denominator: $x - 4 = 0 \implies x = 4$

So, the critical points are $x = -5$ and $x = 4$. These points divide the number line into three intervals:

• $(-\infty, -5)$
• $(-5, 4)$
• $(4, \infty)$

Step 2: Test intervals

Now, test points from each interval to determine the sign of the expression $\frac{x + 5}{x - 4}$.

• For $x \in (-\infty, -5)$, choose $x = -6$: $\frac{-6 + 5}{-6 - 4} = \frac{-1}{-10} = \frac{1}{10} > 0$ (positive)

• For $x \in (-5, 4)$, choose $x = 0$: $\frac{0 + 5}{0 - 4} = \frac{5}{-4} = -\frac{5}{4} < 0$ (negative)

• For $x \in (4, \infty)$, choose $x = 5$: $\frac{5 + 5}{5 - 4} = \frac{10}{1} = 10 > 0$ (positive)

Step 3: Conclusion

The expression is negative or zero in the interval $[-5, 4)$. Since the inequality allows for zero (due to $\leq 0$), the solution includes $x = -5$ but excludes $x = 4$. Therefore, the solution is: $-5 \leq x < 4$

Summary of Solutions:

1. $1 < x < 7$
2. $-5 \leq x < 4$

Would you like further clarification on any part of this process?

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Related questions:

1. How do we solve rational inequalities with higher-degree polynomials?
2. What is the importance of testing intervals when solving inequalities?
3. Can these techniques be applied to systems of inequalities?
4. How do we solve inequalities involving absolute values?
5. What are some real-life applications of solving inequalities?