Question

The solution set of the rational inequality $\frac{x+9}{x-6}\le0$ is

• A. $(-\infty,9)\cup(6,\infty)$
• B. $(-\infty,9]\cup(6,\infty)$
• C. $(-\infty,9]\cup[6,\infty)$
• D. [-9,6)
• E. (-9,6]

Hint:

Inequality Tip: For rational inequalities $\frac{A}{B} \le 0$, always use a hard bracket "[]" for the numerator's zero, and always use a soft parenthesis "()" for the denominator's zero.

Answer 1

The Correct Option is D

Concept:
To solve a rational inequality, find the critical points where the numerator is zero or the denominator is undefined. Use these points to divide the number line into test intervals. Ensure the denominator's root is never included (open bracket) to avoid division by zero.

Step 1: Find the critical points.
Set the numerator and denominator to zero to find the critical boundaries: Numerator: $x + 9 = 0 \implies x = -9$ Denominator: $x - 6 = 0 \implies x = 6$

Step 2: Establish the domain constraint.
Because division by zero is undefined, $x$ cannot mathematically equal 6. Therefore, the critical point $x = 6$ must have an open parenthesis "()" in the final interval notation.

Step 3: Test the intervals on the number line.
The critical points divide the number line into three distinct regions: $x < -9$, $-9 < x < 6$, and $x > 6$. Test a point in the middle region, such as $x = 0$: $$\frac{0 + 9}{0 - 6} = \frac{9}{-6} = -1.5$$ Since $-1.5 \le 0$, the middle region completely satisfies the inequality.

Step 4: Test the boundary conditions.
Because the inequality is "$\le$" (less than or equal to), we must include points that make the expression exactly zero. At $x = -9$, the expression is $0 / -15 = 0$, which is valid. So $-9$ uses a closed bracket "[]".

Step 5: Write the final interval notation.
Combine the valid region (between $-9$ and $6$) with the proper boundary brackets: Closed at $-9$ and open at $6$. Solution set: $[-9, 6)$ Hence the correct answer is (D) [-9,6).