Giải các bất phương trình sau:
a) \((x + 1)(2x - 1) + x \le 3 + 2{x^2}\)
b) \((x + 1)(x + 2)(x + 3) - x > {x^3} + 6{x^2} - 5\)
c) \(x + \sqrt x > (2\sqrt x + 3)(\sqrt x - 1)\)
d) \((\sqrt {1 - x} + 3)(2\sqrt {1 - x} - 5) > \sqrt {1 - x} - 3\)
Gợi ý làm bài
a)
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\(\eqalign{
& (x + 1)(2x - 1) + x \le 3 + 2x_{}^2 \cr
& \Leftrightarrow 2x_{}^2 + 2x - 1 \le 3 + 2x_{}^2 \cr
& \Leftrightarrow 2x \le 4 \Leftrightarrow x \le 2 \cr} \)
b)
\(\eqalign{
& (x + 1)(x + 2)(x + 3) - x > x_{}^3 + 6x_{}^2 - 5 \cr
& \Leftrightarrow x_{}^3 + 6x_{}^2 + 10x + 6 > x_{}^3 + 6x_{}^2 - 5 \cr
& \Leftrightarrow 10x > - 11 \Leftrightarrow x > 1,1 \cr} \)
c)
\(\eqalign{
& x + \sqrt x > (2\sqrt x + 3)(\sqrt x - 1) \cr
& \Leftrightarrow \left\{ \matrix{
x \ge 0 \hfill \cr
x + \sqrt x > 2x + \sqrt x - 3 \hfill \cr} \right. \cr
& \Leftrightarrow \left\{ \matrix{
x \ge 0 \hfill \cr
3 > x \hfill \cr} \right. \Leftrightarrow 0 \le x < 3 \cr} \)
d)
\(\eqalign{
& (\sqrt {1 - x} + 3)(2\sqrt {1 - x} - 5) > \sqrt {1 - x} - 3 \cr
& \Leftrightarrow \left\{ \matrix{
x \le 1 \hfill \cr
2(1 - x) + \sqrt {1 - x} - 15 > \sqrt {1 - x} - 3 \hfill \cr} \right. \cr
& \Leftrightarrow \left\{ \matrix{
x \le 1 \hfill \cr
x < - 5 \hfill \cr} \right. \Leftrightarrow x < - 5 \cr} \)