Solve each of the following system of inequations in R

2(x – 6) < 3x – 7, 11 – 2x < 6 – x

Given 2(x – 6) < 3x – 7 and 11 – 2x < 6 – x

Let us consider the first inequality.

2(x – 6) < 3x – 7

⇒ 2x – 12 < 3x – 7

⇒ 2x – 12 + 12 < 3x – 7 + 12

⇒ 2x < 3x + 5

⇒ 3x + 5 > 2x

⇒ 3x + 5 – 5 > 2x – 5

⇒ 3x > 2x – 5

⇒ 3x – 2x > 2x – 5 – 2x

⇒ x > –5

∴ x ∈ (–5, ∞) (1)

Now, let us consider the second inequality.

11 – 2x < 6 – x

⇒ 11 – 2x – 11 < 6 – x – 11

⇒ –2x < –x – 5

⇒ –x – 5 > –2x

⇒ –x – 5 + 5 > –2x + 5

⇒ –x > –2x + 5

⇒ –x + 2x > –2x + 5 + 2x

⇒ x > 5

∴ x ∈ (5, ∞) (2)

From (1) and (2), we get

x ∈ (–5, ∞) ∩ (5, ∞)

∴ x ∈ (5, ∞)

Thus, the solution of the given system of inequations is (5, ∞).