Solve each of the following system of inequations in R 2x – 7 > 5 �

1.	Solve each of the following system of inequations in R 2x – 7 > 5 – x, 11 – 5x ≤ 1
Answer» Given, 2x – 7 > 5 – x and 11 – 5x ≤ 1 Let us consider the first inequality. 2x – 7 > 5 – x ⇒ 2x – 7 + 7 > 5 – x + 7 ⇒ 2x > 12 – x ⇒ 2x + x > 12 – x + x ⇒ 3x > 12 ⇒ \(\frac{3x}{3}\) > \(\frac{12}{3}\) ⇒ x > 4 ∴ x ∈ ( 4, ∞) ...(1) Now, Let us consider the second inequality. 11 – 5x ≤ 1 ⇒ 11 – 5x – 11 ≤ 1 – 11 ⇒ –5x ≤ –10 ⇒ \(\frac{-5x}{5}\) ≤ \(\frac{-10}{5}\) ⇒ –x ≤ –2 ⇒ x ≥ 2 ∴ x ∈ (2, ∞) ....(2) From (1) and (2), we get x ∈ (4, ∞) ∩ (2, ∞) ∴ x ∈ (4, ∞) Thus, The solution of the given system of inequations is (4, ∞).

Solve each of the following system of inequations in R 2x – 7 > 5 – x, 11 – 5x ≤ 1

Answer»

Given,

2x – 7 > 5 – x and 11 – 5x ≤ 1

Let us consider the first inequality.

2x – 7 > 5 – x

⇒ 2x – 7 + 7 > 5 – x + 7

⇒ 2x > 12 – x

⇒ 2x + x > 12 – x + x

⇒ 3x > 12

⇒ \(\frac{3x}{3}\) > \(\frac{12}{3}\)

⇒ x > 4

∴ x ∈ ( 4, ∞) ...(1)

Now,

Let us consider the second inequality.

11 – 5x ≤ 1

⇒ 11 – 5x – 11 ≤ 1 – 11

⇒ –5x ≤ –10

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

⇒ –x ≤ –2

⇒ x ≥ 2

∴ x ∈ (2, ∞) ....(2)

From (1) and (2), we get

x ∈ (4, ∞) ∩ (2, ∞)

∴ x ∈ (4, ∞)

Thus,

The solution of the given system of inequations is (4, ∞).