Bài 67 trang 151 SGK Đại số 10 nâng cao
Giải các bất phương trình: ...
Giải các bất phương trình:
Giải các bất phương trình:
a) (sqrt {{x^2} + x - 6} < x - 1)
b) (sqrt {2x - 1} le 2x - 3)
c) (sqrt {2{x^2} - 1} > 1 - x)
d) (sqrt {{x^2} - 5x - 14} ge 2x - 1)
Đáp án
a) Ta có:
(eqalign{
& sqrt {{x^2} + x - 6} < x - 1cr& Leftrightarrow left{ matrix{
{x^2} + x - 6 ge 0 hfill cr
x - 1 > 0 hfill cr
{x^2} + x - 6 < {(x - 1)^2} hfill cr}
ight. cr
& Leftrightarrow left{ matrix{
left[ matrix{
x le 3 hfill cr
x ge 2 hfill cr}
ight. hfill cr
x > 1 hfill cr
3x < 7 hfill cr}
ight. Leftrightarrow 2 le x < {7 over 3} cr} )
Vậy (S = { m{[}}2,{7 over 3}))
b) Ta có:
(eqalign{
& sqrt {2x - 1} le 2x - 3 Leftrightarrow left{ matrix{
2x - 1 ge 0 hfill cr
2x - 3 ge 0 hfill cr
2x - 1 le {(2x - 3)^2} hfill cr}
ight. cr
& Leftrightarrow left{ matrix{
x ge {1 over 2} hfill cr
x ge {3 over 2} hfill cr
4{x^2} - 14x + 10 ge 0 hfill cr}
ight.cr& Leftrightarrow left{ matrix{
x ge {3 over 2} hfill cr
left[ matrix{
x le 1 hfill cr
x ge {5 over 2} hfill cr}
ight. hfill cr}
ight. Leftrightarrow x ge {5 over 2} cr} )
Vậy (S = { m{[}}{5 over 2}; + infty ))
c) Ta có:
(eqalign{
& sqrt {2{x^2} - 1} > 1 - x Leftrightarrow left[ matrix{
left{ matrix{
1 - x < 0 hfill cr
2{x^2} - 1 > 0 hfill cr}
ight. hfill cr
left{ matrix{
1 - x ge 0 hfill cr
2{x^2} - 1 > {(1 - x)^2} hfill cr}
ight. hfill cr}
ight. cr
& Leftrightarrow left[ matrix{
x > 1 hfill cr
left{ matrix{
x le 1 hfill cr
{x^2} + 2x - 2 > 0 hfill cr}
ight. hfill cr}
ight. Leftrightarrow left{ matrix{
x > 1 hfill cr
left{ matrix{
x le 1 hfill cr
left[ matrix{
x < - 1 - sqrt 3 hfill cr
x > - 1 + sqrt 3 hfill cr}
ight. hfill cr}
ight. hfill cr}
ight.cr& Leftrightarrow left[ matrix{
x < - 1 - sqrt 3 hfill cr
x > - 1 + sqrt 3 hfill cr}
ight. cr} )
Vậy (S = ( - infty , - 1 - sqrt 3 ) cup ( - 1 + sqrt 3 , + infty ))
d) Ta có:
(eqalign{
& sqrt {{x^2} - 5x - 14} ge 2x - 1 cr
& Leftrightarrow left[ matrix{
left{ matrix{
2x - 1 < 0 hfill cr
{x^2} - 5x - 14 ge 0 hfill cr}
ight. hfill cr
left{ matrix{
2x - 1 ge 0 hfill cr
{x^2} - 5x - 14 ge {(2x - 1)^2} hfill cr}
ight. hfill cr}
ight.cr& Leftrightarrow left[ matrix{
left{ matrix{
x < {1 over 2} hfill cr
left[ matrix{
x le - 2 hfill cr
x ge 7 hfill cr}
ight. hfill cr}
ight. hfill cr
left{ matrix{
x ge {1 over 2} hfill cr
3{x^2} + x + 15 le 0 hfill cr}
ight. hfill cr}
ight. Leftrightarrow x le - 2 cr} )
Vậy (S = (-∞, -2])
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