Solve each of the following system of inequations in R

Given 5x – 7 < 3(x + 3) and

Let us consider the first inequality.

5x – 7 < 3(x + 3)

⇒ 5x – 7 < 3x + 9

⇒ 5x – 7 + 7 < 3x + 9 + 7

⇒ 5x < 3x + 16

⇒ 5x – 3x < 3x + 16 – 3x

⇒ 2x < 16

⇒ x < 8

∴ x ∈ (–∞, 8) (1)

Now, let us consider the second inequality.

⇒ 2 – 3x ≥ 2(x – 4)

⇒ 2 – 3x ≥ 2x – 8

⇒ 2 – 3x – 2 ≥ 2x – 8 – 2

⇒ –3x ≥ 2x – 10

⇒ 2x – 10 ≤ –3x

⇒ 2x – 10 + 10 ≤ –3x + 10

⇒ 2x ≤ –3x + 10

⇒ 2x + 3x ≤ –6x + 10 + 6x

⇒ 5x ≤ 10

⇒ x ≤ 2

∴ x ∈ (–∞, 2] (2)

From (1) and (2), we get

x ∈ (–∞, 8) ∩ (–∞, 2]

∴ x ∈ (–∞, 2]

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

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