Solve each of the following system of inequations in R

x – 2 > 0, 3x < 18

Given x – 2 > 0 and 3x < 18

Let us consider the first inequality.

x – 2 < 0

⇒ x – 2 + 2 < 0 + 2

⇒ x < 2

∴ x ∈ ( 2, ∞) (1)

Now, let us consider the second inequality.

3x < 18

⇒ x < 6

∴ x ∈ (–∞, 6) (2)

From (1) and (2), we get

x ∈ (2, ∞) ∩ (–∞, 6)

∴ x ∈ ( 2, 6)

Thus, the solution of the given system of inequations is (2, 6).