Solve the following linear inequations in R 4x – 2 < 8, when i. x

1.	Solve the following linear inequations in R 4x – 2 < 8, when i. x ∈ R ii. x ∈ Z iii. x ∈ N
Answer» Given, 4x – 2 < 8 ⇒ 4x – 2 + 2 < 8 + 2 ⇒ 4x < 10 ⇒ \(\frac{4x}{4}\) < \(\frac{10}{4}\) ∴ x < \(\frac{5}{2}\) i. x ∈ R When x is a real number, the solution of the given inequation is (-∞,\(\frac{5}{2}\)). ii. x ∈ Z As 2<\(\frac{5}{2}\)<3, When x is an integer, The maximum possible value of x is 2. Thus, The solution of the given inequation is {…, –2, –1, 0, 1, 2}. iii. x ∈ N As 2<\(\frac{5}{2}\)<3, When x is a natural number, the maximum possible value of x is 2 and we know the natural numbers start from 1. Thus, The solution of the given inequation is {1, 2}.

Solve the following linear inequations in R 4x – 2 < 8, when i. x ∈ R ii. x ∈ Z iii. x ∈ N

Answer»

Given,

4x – 2 < 8

⇒ 4x – 2 + 2 < 8 + 2

⇒ 4x < 10

⇒ \(\frac{4x}{4}\) < \(\frac{10}{4}\)

∴ x < \(\frac{5}{2}\)

i. x ∈ R

When x is a real number, the solution of the given inequation is (-∞,\(\frac{5}{2}\)).

ii. x ∈ Z

As 2<\(\frac{5}{2}\)<3,

When x is an integer,

The maximum possible value of x is 2.

Thus,

The solution of the given inequation is {…, –2, –1, 0, 1, 2}.

iii. x ∈ N

As 2<\(\frac{5}{2}\)<3,

When x is a natural number, the maximum possible value of x is 2 and we know the natural numbers start from 1.

Thus,

The solution of the given inequation is {1, 2}.