Solve each of the following system of inequations in R

–5 < 2x – 3 < 5

Given –5 < 2x – 3 < 5

The above inequality can be split into two inequalities.

–5 < 2x – 3 and 2x – 3 < 5

Let us consider the first inequality.

–5 < 2x – 3

⇒ 2x – 3 > –5

⇒ 2x – 3 + 3 > –5 + 3

⇒ 2x > –2

⇒ x > –1

∴ x ∈ (–1, ∞) (1)

Now, let us consider the second inequality.

2x – 3 < 5

⇒ 2x – 3 + 3 < 5 + 3

⇒ 2x < 8

⇒ x < 4

⇒ x > –2

∴ x ∈ (–∞, 4) (2)

From (1) and (2), we get

x ∈ (–1, ∞) ∩ (–∞, 4)

∴ x ∈ (–1, 4)

Thus, the solution of the given system of inequations is (–1, 4).

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