State True or False. If the solution set of the equation $$\sqrt{x + 3 - 4\sqrt{x - 1}} + \sqrt{x + 8 - 6\sqrt{x - 1}} = 1$$ is $$[a, b]$$ then $$b - a = 5$$

Correct Answer: True

Solution

Let $$t = \sqrt{x - 1} \ge 0$$, so $$x = t^2 + 1$$. Then $$x + 3 - 4t = (t - 2)^2$$ and $$x + 8 - 6t = (t - 3)^2$$, and the equation becomes $$|t - 2| + |t - 3| = 1$$, which holds exactly for $$2 \le t \le 3$$. This gives $$5 \le x \le 10$$, so $$b - a = 10 - 5 = 5$$. The statement is True.

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