Solve each of the following system of inequations in R 10 ≤ –5(x

1.	Solve each of the following system of inequations in R 10 ≤ –5(x – 2) < 20
Answer» 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 ⇒ \(\frac{-5x}{5}\) < \(\frac{10}{5}\) ⇒ –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].

Solve each of the following system of inequations in R 10 ≤ –5(x – 2) < 20

Answer»

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

⇒ \(\frac{-5x}{5}\) < \(\frac{10}{5}\)

⇒ –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].