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

1.	Solve each of the following system of inequations in R – 5 < 2x – 3 < 5
Answer» 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 ⇒ \(\frac{2x}{2}\) > \(\frac{-2}{2}\) ⇒ x > –1 ∴ x ∈ (–1, ∞) ....(1) Now, Let us consider the second inequality. 2x – 3 < 5 ⇒ 2x – 3 + 3 < 5 + 3 ⇒ 2x < 8 ⇒ \(\frac{2x}{2}\) < \(\frac{8}{2}\) ⇒ 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).

Solve each of the following system of inequations in R – 5 < 2x – 3 < 5

Answer»

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

⇒ \(\frac{2x}{2}\) > \(\frac{-2}{2}\)

⇒ x > –1

∴ x ∈ (–1, ∞) ....(1)

Now,

Let us consider the second inequality.

2x – 3 < 5

⇒ 2x – 3 + 3 < 5 + 3

⇒ 2x < 8

⇒ \(\frac{2x}{2}\) < \(\frac{8}{2}\)

⇒ 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).