Bài 18 trang 223 Đại số 10 Nâng cao: Giải các bất phương trình...

Giải các bất phương trình

a) 3x² – |5x + 2| >0

b) (sqrt {2{x^2} + 7x + 5}  > x + 1)

c) (sqrt {{x^2} + 4x – 5}  le x + 3)

Đáp án

a) Ta có:

(eqalign{
& 3{x^2} – left| {5x + 2} ight| > 0 Leftrightarrow |5x + 2| < 3{x^2} cr
& Leftrightarrow – 3{x^2} < 5x + 2 < 3{x^2} cr
& Leftrightarrow left{ matrix{
3{x^2} + 5x + 2 > 0 hfill cr
3{x^2} – 5x – 2 > 0 hfill cr} ight. Leftrightarrow left{ matrix{
left[ matrix{
x < – 1 hfill cr
x > – {2 over 3} hfill cr} ight. hfill cr
left{ matrix{
x < – {1 over 3} hfill cr
x > 2 hfill cr} ight. hfill cr} ight. cr&Leftrightarrow left[ matrix{
x < – 1 hfill cr
– {2 over 3} < x < – {1 over 3} hfill cr
x > 2 hfill cr} ight. cr} )

Vậy: (S = ( – infty ,, – 1) cup ( – {2 over 3}; – {1 over 3}) cup (2, + infty ))

b) Ta có:

(eqalign{
& sqrt {2{x^2} + 7x + 5} > x + 1 cr
& Leftrightarrow ,,left[ matrix{
(I),left{ matrix{
x + 1 < 0 hfill cr
2{x^2} + 7x + 5 ge 0 hfill cr} ight. hfill cr
(II)left{ matrix{
x + 1 ge 0 hfill cr
2{x^2} + 7x + 5 > {(x + 1)^2} hfill cr} ight., hfill cr} ight. cr} ) 

Ta có:

((I) Leftrightarrow left{ matrix{
x < – 1 hfill cr
left[ matrix{
x le – {5 over 2} hfill cr
x ge – 1 hfill cr} ight. hfill cr} ight. Leftrightarrow x le – {5 over 2}) 

((II) Leftrightarrow left{ matrix{
x ge – 1 hfill cr
{x^2} + 5x + 4 > 0 hfill cr} ight. )

(Leftrightarrow left{ matrix{
x ge – 1 hfill cr
left[ matrix{
x < – 4 hfill cr
x > – 1 hfill cr} ight. hfill cr} ight. Leftrightarrow x > – 1) 

Vậy: (S = ( – infty ;, – {5 over 2}{ m{]}}, cup ( – 1;, + infty ))

c) Ta có:

(eqalign{
& sqrt {{x^2} + 4x – 5} le x + 3 cr&Leftrightarrow left{ matrix{
x + 3 ge 0 hfill cr
{x^2} + 4x – 5 ge 0 hfill cr
{x^2} + 4x – 5 le {(x + 3)^2} hfill cr} ight. cr
& Leftrightarrow left{ matrix{
x ge – 3 hfill cr
left[ matrix{
x le – 5 hfill cr
x ge 1 hfill cr} ight. hfill cr
x ge – 7 hfill cr} ight. Leftrightarrow x ge 1 cr} )

Vậy (S = [1, +∞))