Question

1) 

a) -2x²+3 ≤ 0

b) -x²- 2x + 3 ≥ 0

c) \(\sqrt{1-3x}\) + x - 2 ≤ 0

Answer 1

a.

\(\Leftrightarrow2x^2\ge3\Leftrightarrow x^2\ge\dfrac{3}{2}\Rightarrow\left[{}\begin{matrix}x\ge\sqrt{\dfrac{3}{2}}\\x\le-\sqrt{\dfrac{3}{2}}\end{matrix}\right.\)

b.

\(\Leftrightarrow\left(1-x\right)\left(x-3\right)\ge0\Rightarrow1\le x\le3\)

c.

\(\Leftrightarrow\sqrt{1-3x}\le2-x\Leftrightarrow\left\{{}\begin{matrix}1-3x\ge0\\2-x\ge0\\1-3x\le x^2-4x+4\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le\dfrac{1}{3}\\x\le2\\x^2-x+3\ge0\end{matrix}\right.\) \(\Leftrightarrow x\le\dfrac{1}{3}\)