MathLabs

Problem 2

Determine all real numbers xx which satisfy 3−x−x+1>12\sqrt{\sqrt{3-x}-\sqrt{x+1}}>\dfrac12.
Step 3 of 5: Set up the equality point
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3−x−x+1=12⟺3−x=14+x+1\sqrt{\sqrt{3-x}-\sqrt{x+1}}=\frac12\Longleftrightarrow\sqrt{3-x}=\frac14+\sqrt{x+1}
Detailed analysis

At the unique boundary point, set f(x)=1/2f(x)=1/2. Squaring the nonnegative outer-root equation once gives 3−x−x+1=1/4\sqrt{3-x}-\sqrt{x+1}=1/4, or equivalently 3−x=1/4+x+1\sqrt{3-x}=1/4+\sqrt{x+1}.