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Chapter 12: Problem 275

Solve the inequality \(\sqrt{(x-3)} \leq 2-\sqrt{(x+1)}\).

Short Answer

Expert verified
The inequality given is \(\sqrt{(x-3)} \leq 2-\sqrt{(x+1)}\). The final inequality after all necessary transformations is \(x^4 - 4x^3 + 8x^2 - 8x + 2 \leq 0\). It is difficult to solve this inequality algebraically, so a graphical or numerical method would be more appropriate. The plot of \(f(x) = x^4 - 4x^3 + 8x^2 - 8x + 2\) will reveal the intervals for which \(f(x) \leq 0\), which are the solutions to the inequality.

Step by step solution

01

Isolate the square root terms

To do this, we will add \(\sqrt{(x+1)}\) to both sides of the inequality: \[\sqrt{(x-3)} + \sqrt{(x+1)} \leq 2\]
02

Square both sides of the inequality

Note that after squaring an inequality, the inequality character might need to be reversed. In this case, it could stay the same as we will square both sides after isolating all the terms. \[(\sqrt{(x-3)} + \sqrt{(x+1)})^2 \leq 2^2\] Expanding both sides, we get: \[(x-3) + 2\sqrt{(x-3)(x+1)} + (x+1) \leq 4\]
03

Simplify the inequality

Combine like terms on the left side of the inequality: \[2x - 2 + 2\sqrt{(x-3)(x+1)} \leq 4\] Subtract 2 from both sides to isolate the term with the square root: \[2\sqrt{(x-3)(x+1)} \leq 2\] Divide both sides by 2: \[\sqrt{(x-3)(x+1)} \leq 1\]
04

Square both sides of the inequality again

Note that after squaring an inequality, the inequality character might need to be reversed. In this case, it could stay the same as we will square both sides after isolating all the terms. \[((x-3)(x+1))^2 \leq 1^2\] Expanding both sides, we get: \[(x^2 - 2x - 3)(x^2 - 2x + 1) \leq 1\]
05

Simplify the inequality

Using the distributive property, multiply the expressions on the left side of the inequality: \[x^4 - 4x^3 + 8x^2 - 8x + 3 \leq 1\] Subtract 1 from both sides of the inequality: \[x^4 - 4x^3 + 8x^2 - 8x + 2 \leq 0\] Unfortunately, this inequality is too difficult to solve algebraically. The initial method we used did not work because squaring both sides has the potential to introduce extraneous solutions. A graphical or numerical method would be more appropriate to find the solution for this inequality. The plot of \(f(x) = x^4 - 4x^3 + 8x^2 - 8x + 2\) will reveal the intervals for which \(f(x) \leq 0\), which are the solutions to the inequality.

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