Solve each of the following system of inequations in R

2x – 7 > 5 – x, 11 – 5x ≤ 1

Given 2x – 7 > 5 – x and 11 – 5x ≤ 1

Let us consider the first inequality.

2x – 7 > 5 – x

⇒ 2x – 7 + 7 > 5 – x + 7

⇒ 2x > 12 – x

⇒ 2x + x > 12 – x + x

⇒ 3x > 12

⇒ x > 4

∴ x ∈ ( 4, ∞) (1)

Now, let us consider the second inequality.

11 – 5x ≤ 1

⇒ 11 – 5x – 11 ≤ 1 – 11

⇒ –5x ≤ –10

⇒ –x ≤ –2

⇒ x ≥ 2

∴ x ∈ (2, ∞) (2)

From (1) and (2), we get

x ∈ (4, ∞) ∩ (2, ∞)

∴ x ∈ (4, ∞)

Thus, the solution of the given system of inequations is (4, ∞).

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