Câu 29 trang 121 SGK Đại số 10 nâng cao

Giải các hệ bất phương trình

Giải các hệ bất phương trình

a) 

(left{ matrix{
{{5x + 2} over 3} ge 4 - x hfill cr
{{6 - 5x} over {13}} < 3x + 1 hfill cr} ight.)

b)

(left{ matrix{
{(1 - x)^2} > 5 + 3x + {x^2} hfill cr
{(x + 2)^3} < {x^3} + 6{x^2} - 7x - 5 hfill cr} ight.)

c)

(left{ matrix{
{{4x - 5} over 7}< x + 3 hfill cr
{{3x + 8} over 4} > 2x - 5 hfill cr} ight.)

d)

(left{ matrix{
x - 1 le 2x - 3 hfill cr
3x < x + 5 hfill cr
{{5 - 3x} over 2} le x - 3 hfill cr} ight.)

Đáp án

a) Ta có:

(eqalign{
& left{ matrix{
{{5x + 2} over 3} ge 4 - x hfill cr
{{6 - 5x} over {13}} < 3x + 1 hfill cr} ight. Leftrightarrow left{ matrix{
5x + 2 ge 12 - 3x hfill cr
6 - 5x < 39x + 13 hfill cr} ight. cr
& Leftrightarrow left{ matrix{
8x ge 10 hfill cr
44x > - 7 hfill cr} ight. Leftrightarrow left{ matrix{
x ge {5 over 4} hfill cr
x > - {7 over {44}} hfill cr} ight. Leftrightarrow x ge {5 over 4} cr} ) 

Vậy (S = { m{[}}{5 over 4}; + infty ))

b) Ta có:

(eqalign{
& left{ matrix{
{(1 - x)^2} > 5 + 3x + {x^2} hfill cr
{(x + 2)^3} < {x^3} + 6{x^2} - 7x - 5 hfill cr} ight. cr&Leftrightarrow left{ matrix{
1 - 2x + {x^2} > 5 + 3x + {x^2} hfill cr
{x^3} + 6{x^2} + 12x + 8 < {x^3} + 6{x^2} - 7x - 5 hfill cr} ight. cr
& Leftrightarrow left{ matrix{
5x < - 4 hfill cr
19x < - 13 hfill cr} ight. Leftrightarrow left{ matrix{
x < - {4 over 5} hfill cr
x < - {{13} over {19}} hfill cr} ight. Leftrightarrow x < - {4 over 5} cr} )

Vậy (S = ( - infty ; - {4 over 5}))

c) Ta có:

(eqalign{
& left{ matrix{
{{4x - 5} over 7} < x + 3 hfill cr
{{3x + 8} over 4} > 2x - 5 hfill cr} ight. Leftrightarrow left{ matrix{
4x - 5 < 7x + 21 hfill cr
3x + 8 > 8x - 20 hfill cr} ight. cr&Leftrightarrow left{ matrix{
3x > - 26 hfill cr
5x < 28 hfill cr} ight. cr
& Leftrightarrow left{ matrix{
x > - {{26} over 3} hfill cr
x < {{28} over 5} hfill cr} ight. Leftrightarrow - {{26} over 3} < x < {{28} over 5} cr} )

Vậy (S = ( - {{26} over 3};{{28} over 5}))

d) Ta có:

(left{ matrix{
x - 1 le 2x - 3 hfill cr
3x < x + 5 hfill cr
{{5 - 3x} over 2} le x - 3 hfill cr} ight. Leftrightarrow left{ matrix{
x ge 2 hfill cr
2x < 5 hfill cr
5 - 3x le 2x - 6 hfill cr} ight. )

(Leftrightarrow left{ matrix{
x ge 2 hfill cr
x < {5 over 2} hfill cr
5x ge 11 hfill cr} ight.Leftrightarrow {{11} over 5} le x <{5 over 2})

Vậy (S = { m{[}}{{11} over 5};{5 over 2}))

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