To solve the inequality $\sqrt{x - 1} - \sqrt{11 - x} \ge 2,$ we'll go through the following steps:

Step 1: Isolate one of the square roots

Move $\sqrt{11 - x}$ to the other side: $\sqrt{x - 1} \ge \sqrt{11 - x} + 2.$

Step 2: Square both sides to eliminate the square roots

To remove the square roots, we square both sides of the inequality: $x - 1 \ge (\sqrt{11 - x} + 2)^2.$ Expanding the right side: $x - 1 \ge (11 - x) + 4\sqrt{11 - x} + 4.$ Simplify: $x - 1 \ge 15 - x + 4\sqrt{11 - x}.$ Rearrange terms to isolate the square root: $2x - 16 \ge 4\sqrt{11 - x}.$ Divide by 4: $\frac{x - 8}{2} \ge \sqrt{11 - x}.$

Step 3: Square both sides again

Square both sides once more to eliminate the square root: $\left(\frac{x - 8}{2}\right)^2 \ge 11 - x.$ Expanding the left side: $\frac{(x - 8)^2}{4} \ge 11 - x.$ Multiply through by 4: $(x - 8)^2 \ge 44 - 4x.$ Expanding $(x - 8)^2$: $x^2 - 16x + 64 \ge 44 - 4x.$ Rearrange terms: $x^2 - 12x + 20 \ge 0.$

Step 4: Solve the resulting quadratic inequality

Now we solve $x^2 - 12x + 20 \ge 0$.

1. Find the roots of the equation $x^2 - 12x + 20 = 0$ using the quadratic formula: $x = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 20}}{2 \cdot 1} = \frac{12 \pm \sqrt{144 - 80}}{2} = \frac{12 \pm \sqrt{64}}{2}.$ Simplify the square root: $x = \frac{12 \pm 8}{2}.$ Thus, the roots are: $x = 10 \quad \text{and} \quad x = 2.$

2. Test intervals around the roots $x = 2$ and $x = 10$ for $x^2 - 12x + 20 \ge 0$.

   • For $x < 2$: Choose $x = 0$. Then $x^2 - 12x + 20 = 20 > 0$.
   • For $2 < x < 10$: Choose $x = 5$. Then $x^2 - 12x + 20 = -15 < 0$.
   • For $x > 10$: Choose $x = 11$. Then $x^2 - 12x + 20 = 23 > 0$.

   Therefore, $x \in (-\infty, 2] \cup [10, \infty)$.

Step 5: Check for extraneous solutions

The original inequality involves square roots, so we need $x - 1 \ge 0$ and $11 - x \ge 0$. This implies: $1 \le x \le 11.$ Thus, we restrict the solution to $x \in [2, 10]$.

Final Answer

The solution is: $x \in [10, 11].$

Would you like further clarification, or any other details?

Related Questions

1. How do we solve inequalities involving square roots and absolute values?
2. What are common methods for checking extraneous solutions?
3. Can all inequalities be solved by squaring both sides?
4. How do domain restrictions affect solutions to inequalities?
5. What are other techniques for isolating terms in inequalities?

Tip: Always check the domain of the function when solving inequalities involving roots or rational expressions.