Correct Answer: C) 18\n\nLet's solve the given inequality step-by-step.\n\nThe inequality is $\\frac{x-1}{x-3} 0$ and when $x-3 0$\n\nStep 5: Identify the critical points. These are the values of $x$ for which the numerator or the denominator becomes zero. These points divide the number line into intervals where the expression's sign remains constant.\nCritical points are $x=5$ (from the numerator $x-5=0$) and $x=3$ (from the denominator $x-3=0$). Note that $x \\neq 3$ since it would make the denominator zero, rendering the expression undefined.\n\nStep 6: Use a number line and test values (or the wavy curve method) to determine the sign of the expression $\\frac{x - 5}{x-3}$ in the intervals defined by the critical points.\n\nWe place the critical points 3 and 5 on a number line, creating three intervals:\n\n* **Interval 1: $x 0$, the inequality $\\frac{x-5}{x-3} > 0$ holds true for all $x$ in $(-\\infty, 3)$.\n\n* **Interval 2: $3 0$ does not hold true for any $x$ in $(3, 5)$.\n\n* **Interval 3: $x > 5$**\n Choose a test value, e.g., $x=6$. Substitute into $\\frac{x-5}{x-3}$: $\\frac{6-5}{6-3} = \\frac{1}{3}$. Since $\\frac{1}{3} > 0$, the inequality $\\frac{x-5}{x-3} > 0$ holds true for all $x$ in $(5, \\infty)$.\n\nStep 7: Combine the intervals where the inequality $\\frac{x - 5}{x-3} > 0$ is satisfied.\nThe solution set for the inequality is $x \\in (-\infty, 3) \\cup (5, \\infty)$.\n\nStep 8: Determine the number of integer values of $x$ within the given range $[-10, 10]$ that satisfy this solution set.\n\nWe need to find integers $x$ such that $x \\in [-10, 10]$ AND ($x 5$).\n\n* **For $x 5$:** The integers in the interval $[-10, 10]$ that are greater than 5 are $\\{6, 7, 8, 9, 10\\}$.\n The number of integers in this range is $10 - 6 + 1 = 5$.\n\nStep 9: Add the counts from both valid intervals.\nTotal number of integer values = $13 + 5 = 18$.\n\nThus, there are 18 integer values of $x$ in the interval $[-10, 10]$ that satisfy the given inequality.\n\nThe final answer is $\\boxed{\\text{18}}$.