Question

Cho mệnh đề chứa biến P(x), với x \(\in R\). Tìm x để P(x) là mệnh đề đúng?

a) P(x): " \(x^2-5x+4=0\) " 

b) P(x): " \(x^2-5x+6=0\) "

c) P(x): " \(x^2-3x>0\) "

d) P(x): "\(\sqrt{x}>x\) "

e) P(x): " 2x+ 3<7 "

f) P(x): " \(x^2+x+1>0\) "

Answer 1

d) \(\sqrt[]{x}>x\)

\(\Leftrightarrow x-\sqrt[]{x}< 0\)

\(\Leftrightarrow\sqrt[]{x}\left(\sqrt[]{x}-1\right)< 0\left(x\ge0\right)\)

\(\Leftrightarrow0< x< 1\)

Answer 2

a) \(P\left(x\right):"x^2-5x+4=0"\)

\(x^2-5x+4=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=4\end{matrix}\right.\)

Vậy \(x\in\left\{1;4\right\}\) để \(P\left(x\right):"x^2-5x+4=0"\) đúng

b) \(P\left(x\right):"x^2-5x+6=0"\)

\(x^2-5x+6=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=3\end{matrix}\right.\)

Vậy \(x\in\left\{2;3\right\}\) để \(P\left(x\right):"x^2-5x+6=0"\) đúng

c) \(P\left(x\right):"x^2-3x=0"\)

\(x^2-3x=0\)

\(\Leftrightarrow x\left(x-3\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=3\end{matrix}\right.\)

Vậy \(x\in\left\{0;3\right\}\) để \(P\left(x\right):"x^2-3x=0"\) đúng

d) \(P\left(x\right):"\sqrt[]{x}>x"\)

\(\sqrt[]{x}>x\)

\(\Leftrightarrow x-\sqrt[]{x}< 0\)

\(\Leftrightarrow\sqrt[]{x}\left(\sqrt[]{x}-1\right)< 0\)

\(\Leftrightarrow0< x< 1\)

Vậy \(x\in\left(0;1\right)\) để \(P\left(x\right):"\sqrt[]{x}>x"\) đúng

e) \(P\left(x\right):"2x+3< 7"\)

\(2x+3< 7\)

\(\Leftrightarrow2x< 4\)

\(\Leftrightarrow x< 2\)

Vậy \(x\in(-\infty;2)\) để \(P\left(x\right):"2x+3< 7"\) đúng

f) \(P\left(x\right):"x^2+x+1>0"\)

\(x^2+x+1>0\)

\(\Leftrightarrow x^2+x+\dfrac{1}{4}+\dfrac{3}{4}>0\)

\(\Leftrightarrow\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\)

\(\Leftrightarrow\forall x\in R\) để \(P\left(x\right):"x^2+x+1>0"\) đúng