Solve each of the following system of inequations in R 3x – 6 > 0,

1.	Solve each of the following system of inequations in R 3x – 6 > 0, 2x – 5 > 0
Answer» Given, 3x – 6 > 0 and 2x – 5 > 0 Let us consider the first inequality. 3x – 6 > 0 ⇒ 3x – 6 + 6 > 0 + 6 ⇒ 3x > 6 ⇒ \(\frac{3x}{3}\) > \(\frac{6}{3}\) ⇒ x > 2 ∴ x ∈ ( 2, ∞) ...(1) Now, Let us consider the second inequality. 2x – 5 > 0 ⇒ 2x – 5 + 5 > 0 + 5 ⇒ 2x > 5 ⇒ \(\frac{2x}{2}\) > \(\frac{5}{2}\) ⇒ x > \(\frac{5}{2}\) ∴ x ∈ (\(\frac{5}{2}\),∞) ...(2) From (1) and (2), we get x ∈ (2,∞) ∩ (\(\frac{5}{2}\),∞) ∴ x ∈ (\(\frac{5}{2}\),∞) Thus, The solution of the given system of inequations is (\(\frac{5}{2}\),∞).

Solve each of the following system of inequations in R 3x – 6 > 0, 2x – 5 > 0

Answer»

Given,

3x – 6 > 0 and 2x – 5 > 0

Let us consider the first inequality.

3x – 6 > 0

⇒ 3x – 6 + 6 > 0 + 6

⇒ 3x > 6

⇒ \(\frac{3x}{3}\) > \(\frac{6}{3}\)

⇒ x > 2

∴ x ∈ ( 2, ∞) ...(1)

Now,

Let us consider the second inequality.

2x – 5 > 0

⇒ 2x – 5 + 5 > 0 + 5

⇒ 2x > 5

⇒ \(\frac{2x}{2}\) > \(\frac{5}{2}\)

⇒ x > \(\frac{5}{2}\)

∴ x ∈ (\(\frac{5}{2}\),∞) ...(2)

From (1) and (2), we get

x ∈ (2,∞) ∩ (\(\frac{5}{2}\),∞)

∴ x ∈ (\(\frac{5}{2}\),∞)

Thus,

The solution of the given system of inequations is (\(\frac{5}{2}\),∞).