Bài 44: Giải các bất phương trình sau:
a) $\({{x + 1} \over {x – 1}} + 2 > {{x – 1} \over x};\)
b) \({1 \over {x + 1}} + {2 \over {x + 3}} < {3 \over {x + 2}}.\)
a) \(\eqalign{
& {{x + 1} \over {x – 1}} + 2 > {{x – 1} \over x} \Leftrightarrow {{3x – 1} \over {x – 1}} > {{x – 1} \over x} \cr
& \Leftrightarrow {{3{x^2} – x – {{(x – 1)}^2}} \over {x(x – 1)}} > 0 \Leftrightarrow {{2{x^2} + x – 1} \over {x(x – 1)}} > 0 \cr} \)
\( \Leftrightarrow x < – 1\) hoặc \(0 < x < {1 \over 2}\) hoặc \(x > 1\)
b) \(\eqalign{
& {1 \over {x + 1}} + {2 \over {x + 3}} + {3 \over {x + 2}} < 0 \cr
& \Leftrightarrow {{x + 3 + 2x + 2} \over {(x + 1)(x + 3)}} < {3 \over {x + 2}} \cr} \)
\( \Leftrightarrow {{(3x + 5)(x + 2) – 3(x + 1) + (x + 3)} \over {(x + 1)(x + 2)(x + 3)}} < 0.\)
\( \Leftrightarrow {{1 – x} \over {(x + 1)(x + 2)(x + 3)}} < 0\)
\( \Leftrightarrow x < – 3\) hoặc \( – 2 < x < – 1\) hoặc \(x > 1\)
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Đáp số: x < -3 hoặc -2 < x < -1 hoặc x > 1
Bài 45: Giải các bất phương trình sau:
a) \(\left\{ \matrix{
{x^2} \ge 0,25 \hfill \cr
{x^2} – x \le 0 \hfill \cr} \right.;\)
b) \(\left\{ \matrix{
(x – 1)(2x + 3) > 0 \hfill \cr
(x – 4)(x + {1 \over 4}) \le 0 \hfill \cr} \right.\)
a) \(\left\{ \matrix{
{x^2} \ge 0,25 \hfill \cr
{x^2} – x \le 0 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
{x^2} – 0,25 \ge 0 \hfill \cr
{x^2} – x \le 0 \hfill \cr} \right. \Leftrightarrow 0,5 \le x \le 1.\)
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b) \(\eqalign{
& \left\{ \matrix{
(x – 1)(2x + 3) > 0 \hfill \cr
(x – 4)(x + {1 \over 4}) \le 0 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
x \in ( – \infty ; – {3 \over 2}) \cup (1; + \infty ) \hfill \cr
x \in {\rm{[ – }}{1 \over 4}{\rm{;4]}} \hfill \cr} \right. \cr
& \Leftrightarrow x \in (1;4]. \cr} \)
Bài 46: Giải các bất phương trình sau:
a) \(\left\{ \matrix{
{x^2} \ge 0,25 \hfill \cr
{x^2} – x \le 0 \hfill \cr} \right.;\)
b) \(\left\{ \matrix{
(x – 1)(2x + 3) > 0 \hfill \cr
(x – 4)(x + {1 \over 4}) \le 0 \hfill \cr} \right.\)
a) \(\eqalign{
& \left\{ \matrix{
{x^2} \ge 4x \hfill \cr
{(2x – 1)^2} < 9 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
{x^2} – 4x \ge 0 \hfill \cr
– 3 < 2x – 1 < 3 \hfill \cr} \right. \cr
& \Leftrightarrow \left\{ \matrix{
x \in ( – \infty ;0] \cup {\rm{[}}4; + \infty ) \hfill \cr
– 1 < x < 2 \hfill \cr} \right. \Leftrightarrow – 1 < x \le 0 \cr} \)
b) \(\eqalign{
& \left\{ \matrix{
2x – 3 < (x + 1)(x – 2) \hfill \cr
{x^2} – x \le 6 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
{x^2} – 3x + 1 > 0 \hfill \cr
{x^2} – x – 6 \le 0 \hfill \cr} \right. \cr
& \Leftrightarrow \left\{ \matrix{
x \in ( – \infty ;{{3 – \sqrt 5 } \over 2}) \cup ({{3 + \sqrt 5 } \over 2};3] \hfill \cr
– 2 \le x \le 3 \hfill \cr} \right. \cr} \)
\( \Leftrightarrow x \in {\rm{[ – 2;}}{{3 – \sqrt 5 } \over 2}) \cup ({{3 + \sqrt 5 } \over 2};3{\rm{]}}\)
Bài 47: Giải các bất phương trình, hệ bất phương trình (ẩn m) sau:
a) \(2{m^2} – m – 5 > 0;\)
b) \( – {m^2} + m + 9 > 0.$\)
a) \(2{m^2} – m – 5 > 0 \Leftrightarrow m < {{1 – \sqrt {41} } \over 4};m > {{1 + \sqrt {41} } \over 4}\)
b) \(- {m^2} + m + 9 > 0 \Leftrightarrow {{1 – \sqrt {37} } \over 2} < m < {{1 + \sqrt {37} } \over 2}\)
