(i) 4n + 7 ≥ 3n + 10, n is an integer.
4n + 7 – 3n ≥ 3n + 10 – 3n
n(4 – 3) + 7 ≥ 3n + 10 – 3n
n(4 – 3) + 7 ≥ n (3 – 3) + 10
n + 7 ≥ 10
Subtracting 7 on both sides
n + 7 – 7 ≥ 10 – 7
n ≥ 3
Since the solution is an integer and is greater than or equal to 3, the solution will be 3,
4, 5, 6, 7, …
n = 3, 4, 5, 6,7, …
(ii) 6(x + 6) ≥ 5(x – 3), x is a whole number.
6x + 36 ≥ 5x – 15
Subtracting 5x on both sides
6x + 36 – 5x ≥ 5x – 15 – 5x
x (6 – 5) + 36 ≥ x(5 – 5) – 15
x + 36 ≥ -15
Subtracting 36 on both sides
x + 36 – 36 ≥ -15 -36
x ≥ -51
The solution is a whole number and which is greater than or equal to -51
∴ The solution is 0, 1, 2, 3, 4,…
x = 0,1,2, 3,4,…
(iii) -13 ≤ 5x + 2 ≤ 32, x is an integer.
Subtracting throughout by 2
-13 – 2 ≤ 5x + 2 – 2 ≤ 32 – 2
-15 ≤ 5x ≤ 30
Dividing throughout by 5
\(\frac{-15}{5}\) ≤ \(\frac{5x}{5}\) ≤ \(\frac{30}{5}\)
– 3 ≤ x ≤ 6
∴ Since the solution is an integer between -3 and 6 both inclusive, we have the solution
As -3, -2, -1,0, 1,2, 3, 4, 5, 6.
i.e. x = -3, -2, 0, 1, 2, 3,4, 5 and 6.
