MathLabs

Problem 2

Determine all real numbers xx which satisfy 3−x−x+1>12\sqrt{\sqrt{3-x}-\sqrt{x+1}}>\dfrac12.
Step 5 of 5: Choose the valid root and state the interval
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x1,2=1±12732⟹x∈[−1,1−12732)x_{1,2}=1\pm\frac{\sqrt{127}}{32}\quad\Longrightarrow\quad\boxed{x\in\left[-1,1-\frac{\sqrt{127}}{32}\right)}
Detailed analysis

The quadratic roots are x1,2=1±127/32x_{1,2}=1\pm\sqrt{127}/32. The larger root is greater than 11 and lies outside the domain; it is introduced by squaring. The equality threshold is therefore 1−127/321-\sqrt{127}/32, and monotonicity gives the solution interval [−1,1−127/32)[-1,1-\sqrt{127}/32).