Solve each of the following system of inequations in R

10 ≤ –5(x – 2) < 20

Given 10 ≤ –5(x – 2) < 20

The above inequality can be split into two inequalities.

10 ≤ –5(x – 2) and –5(x – 2) < 20

Let us consider the first inequality.

10 ≤ –5(x – 2)

⇒ 10 ≤ –5x + 10

⇒ 10 – 10 ≤ –5x + 10 – 10

⇒ 0 ≤ –5x

⇒ 0 + 5x ≤ –5x + 5x

⇒ 5x ≤ 0

⇒ x ≤ 0

∴ x ∈ (–∞, 0] (1)

Now, let us consider the second inequality.

–5(x – 2) < 20

⇒ –5x + 10 < 20

⇒ –5x + 10 – 10 < 20 – 10

⇒ –5x < 10

⇒ –x < 2

⇒ x > –2

∴ x ∈ (–2, ∞) (2)

From (1) and (2), we get

x ∈ (–∞, 0] ∩ (–2, ∞)

∴ x ∈ (–2, 0]

Thus, the solution of the given system of inequations is (–2, 0].

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