What is the solution of $\frac{x - 5}{x + 3} < -1$?

Analysis

To solve the inequality $\frac{x - 5}{x + 3} < -1$, we must move all terms to one side to compare the expression to zero. We cannot simply multiply both sides by $(x + 3)$ because we do not know if $(x + 3)$ is positive or negative, and multiplying by a negative number would flip the inequality sign.

Concept

When solving rational inequalities:

1. Move all terms to one side to get $f(x) < 0$.
2. Find the critical values (where the numerator is zero and where the denominator is zero).
3. Use a sign chart or test intervals to determine where the expression is negative.

Step-by-Step Solution

Step 1: Move -1 to the left side.

Step 2: Find a common denominator.

Step 3: Identify critical values.

• Numerator: $2x - 2 = 0 \implies x = 1$
• Denominator: $x + 3 = 0 \implies x = -3$

Step 4: Test the intervals. Our critical values divide the number line into three regions: $x < -3$, $-3 < x < 1$, and $x > 1$.

• Test $x = -4$ (Region $x < -3$): $\frac{2(-4) - 2}{-4 + 3} = \frac{-10}{-1} = 10$. (Positive, so this is not the solution).
• Test $x = 0$ (Region $-3 < x < 1$): $\frac{2(0) - 2}{0 + 3} = \frac{-2}{3}$. (Negative, so this is the solution).
• Test $x = 2$ (Region $x > 1$): $\frac{2(2) - 2}{2 + 3} = \frac{2}{5}$. (Positive, so this is not the solution).

The inequality holds true when $-3 < x < 1$.

Conclusion

Comparing our result to the given options, the correct range is $-3 < x < 1$. This matches option A.

Exam Tip

When dealing with rational inequalities, never cross-multiply unless you are absolutely certain of the sign of the denominator. Always bring everything to one side and set it to zero; this prevents the common mistake of forgetting to flip the inequality sign!