We need to solve the inequality:

$\frac{x+5}{x-3} \geq 0$

Step 1: Find Critical Points

The expression is zero or undefined at:

1. Numerator: $x + 5 = 0 \Rightarrow x = -5$ (makes the fraction zero)
2. Denominator: $x - 3 = 0 \Rightarrow x = 3$ (makes the fraction undefined)

These values split the number line into intervals.

Step 2: Test Intervals

The critical points divide the number line into three intervals:

1. $(-\infty, -5)$
2. $(-5, 3)$
3. $(3, \infty)$

We test a number in each interval.

• For $x = -6$ in $(-\infty, -5)$: $\frac{-6+5}{-6-3} = \frac{-1}{-9} = \frac{1}{9} > 0$ ✅ Positive

• For $x = 0$ in $(-5, 3)$: $\frac{0+5}{0-3} = \frac{5}{-3} < 0$ ❌ Negative

• For $x = 4$ in $(3, \infty)$: $\frac{4+5}{4-3} = \frac{9}{1} = 9 > 0$ ✅ Positive

Step 3: Include or Exclude Critical Points

• $x = -5$ makes the fraction zero, so we include $x = -5$.
• $x = 3$ makes the fraction undefined, so we exclude $x = 3$.

Step 4: Write the Solution

The solution includes the intervals where the fraction is non-negative:

$(-\infty, -5] \cup (3, \infty)$

Final Answer:

$(-\infty, -5] \cup (3, \infty)$

Would you like a graph to visualize this?

Related Questions:

1. How do we solve rational inequalities in general?
2. What happens if the inequality were strict ($>$ instead of $\geq$)?
3. How would we solve $\frac{x-2}{x+1} < 0$?
4. How does the sign test work for polynomial inequalities?
5. How can we check the solution graphically?

Tip:

Always exclude points where the denominator is zero when solving rational inequalities!