Solve the following inequality: \[ \frac{x-1}{x^{2}-x-6} \geq 0 \] Need to include all the key features as well as sketch of either a graph or a number line.

Step 1 ：Understand the problem: We are asked to solve the inequality \(\frac{x-1}{x^{2}-x-6} \geq 0\) and provide a sketch of either a graph or a number line.

Step 2 ：Factor the denominator of the fraction. The quadratic equation \(x^{2}-x-6\) can be factored into \((x-3)(x+2)\). So, the inequality becomes \(\frac{x-1}{(x-3)(x+2)} \geq 0\).

Step 3 ：Find the critical points by setting the numerator and the denominator equal to zero and solving for x. For the numerator, \(x-1=0\), we get \(x=1\). For the denominator, \((x-3)(x+2)=0\), we get \(x=3\) and \(x=-2\).

Step 4 ：Test the intervals: \((-∞, -2)\), \((-2, 1)\), \((1, 3)\), and \((3, ∞)\). Choose a test point in each interval and substitute it into the inequality. If the inequality is true, the interval is part of the solution. If the inequality is false, the interval is not part of the solution.

Step 5 ：For \((-∞, -2)\), choose \(x=-3\). Substituting \(x=-3\) into the inequality gives \(\frac{-3-1}{(-3-3)(-3+2)} = \frac{-4}{6} = -\frac{2}{3} < 0\). So, \((-∞, -2)\) is not part of the solution.

Step 6 ：For \((-2, 1)\), choose \(x=0\). Substituting \(x=0\) into the inequality gives \(\frac{0-1}{(0-3)(0+2)} = \frac{-1}{6} = \frac{1}{6} > 0\). So, \((-2, 1)\) is part of the solution.

Step 7 ：For \((1, 3)\), choose \(x=2\). Substituting \(x=2\) into the inequality gives \(\frac{2-1}{(2-3)(2+2)} = \frac{1}{4} = -\frac{1}{4} < 0\). So, \((1, 3)\) is not part of the solution.

Step 8 ：For \((3, ∞)\), choose \(x=4\). Substituting \(x=4\) into the inequality gives \(\frac{4-1}{(4-3)(4+2)} = \frac{3}{6} = \frac{1}{2} > 0\). So, \((3, ∞)\) is part of the solution.

Step 9 ：The solution to the inequality is \((-2, 1] ∪ (3, ∞)\).

Step 10 ：Check the solution: Substitute a number from each interval into the inequality to verify that the solution is correct. For example, for the interval \((-2, 1]\), choose \(x=0\), and for the interval \((3, ∞)\), choose \(x=4\). Both of these values satisfy the inequality, so the solution is correct.

Step 11 ：Sketch the graph or number line: On a number line, we would mark -2 and 1 with open circles (since these values are not included in the solution) and 3 with a closed circle (since this value is included in the solution). We would then shade the interval between -2 and 1 and the interval from 3 to infinity.