Question

11. Solve the following inequalities.
(i) 4n + 7 >_ 3n + 10, n is an integer.
(ii) 6(x + 6) >_ 5(x - 3), x is a whole number
(iii) -13 <_ 5x + 2 <_32, x is an integer.​

Answer 1

i) 4n + 7 >_ 3n + 10, n is an integer.

Solution:

/* Subtract 7 ,both sides of inequality */

=> 4n + 7 - 7 ≥ 3n + 10 - 7

=> 4n ≥ 3n + 3

/* Subtract 3n , both sides of the inequality */

=> 4n - 3n ≥ 3n - 3n + 3

=> n ≥ 3

Therefore.,

$n = \{ 3,4,5,6,\cdot\cdot\cdot , \}$

ii) 6(x + 6) >_ 5(x - 3), x is a whole number

Solution:

=> 6x + 36 ≥ 5x - 15

/* subtract bothsides by 36 , we get */

=> 6x + 36 - 36 ≥ 5x - 15 - 36

=> 6x ≥ 5x - 51

/* Subtract bothsides by 5x , we get */

=> 6x - 5x ≥ 5x - 5x - 51

=> x ≥ -51

/* But x is a whole number */

Therefore.,

$Value \:of \: x = \{ 0,1,2,3,,\cdot\cdot\cdot , \}$

(iii) -13 <_ 5x + 2 <_32, x is an integer.

=> (-13 - 2) ≤ (5x+2-2) ≤ (32-2)

=> -15 ≤ 5x ≤ 30

/* Divide each term by 5 ,we get */

=> -3 ≤ x ≤ 6

/* But x is an integer */

Therefore.,

$Value \: of \: x = \{ -3,-2,-1,0,1,2,3,4,5,6 \}$

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