Solve each of the following system of inequations in R

Given and

Let us consider the first inequality.

For this inequation to be true, there are two possible cases.

i. 11x – 2 > 0 and 7x – 1 < 0

⇒ 11x – 2 + 2 > 0 + 2 and 7x – 1 + 1 < 0 + 1

⇒ 11x > 2 and 7x < 1

However,

Hence, this case is not possible.

ii. 11x – 2 < 0 and 7x – 1 > 0

⇒ 11x – 2 + 2 < 0 + 2 and 7x – 1 + 1 > 0 + 1

⇒ 11x < 2 and 7x > 1

However,

Hence, (1)

Now, let us consider the second inequality.

For this inequation to be true, there are two possible cases.

i. x – 23 > 0 and x – 8 < 0

⇒ x – 23 + 23 > 0 + 23 and x – 8 + 8 < 0 + 8

⇒ x > 23 and x < 8

∴ x ∈ (23, ∞) ∩ (–∞, 5)

However, (23, ∞) ∩ (–∞, 5) = ∅

Hence, this case is not possible.

ii. x – 23 < 0 and x – 8 > 0

⇒ x – 23 + 23 < 0 + 23 and x – 8 + 8 > 0 + 8

⇒ x < 23 and x > 8

∴ x ∈ (–∞, 23) ∩ (8, ∞)

However, (–∞, 23) ∩ (8, ∞) = (8, 23)

Hence, x ∈ (8, 23) (2)

From (1) and (2), we get

∴ x ∈ ∅

Thus, there is no solution of the given system of inequations.

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