Solve each of the following system of inequations in R 4x – 1 ≤ 0

1.	Solve each of the following system of inequations in R 4x – 1 ≤ 0, 3 – 4x < 0
Answer» Given, 4x – 1 ≤ 0 and 3 – 4x < 0 Let us consider the first inequality. 4x – 1 ≤ 0 ⇒ 4x – 1 + 1 ≤ 0 + 1 ⇒ 4x ≤ 1 ⇒ \(\frac{4x}{4}\) ≤ \(\frac{1}{4}\) ⇒ x ≤ \(\frac{1}{4}\) ∴ x ∈ ( -∞,\(\frac{1}{4}\)] ... (1) Now, Let us consider the second inequality. 3 – 4x < 0 ⇒ 3 – 4x – 3 < 0 – 3 ⇒ –4x < –3 ⇒ \(\frac{-4x}{4}\) < \(\frac{-3}{4}\) ⇒ -x < \(-\frac{3}{4}\) ⇒ x > \(\frac{3}{4}\) ∴ x ∈ (\(\frac{3}{4}\),∞) ... (2) From (1) and (2), we get x ∈ ( -∞,\(\frac{1}{4}\)] ∩ (\(\frac{3}{4}\),∞) ∴ x ∈ ∅ Thus, There is no solution of the given system of inequations.

Solve each of the following system of inequations in R 4x – 1 ≤ 0, 3 – 4x < 0

Answer»

Given,

4x – 1 ≤ 0 and 3 – 4x < 0

Let us consider the first inequality.

4x – 1 ≤ 0

⇒ 4x – 1 + 1 ≤ 0 + 1

⇒ 4x ≤ 1

⇒ \(\frac{4x}{4}\) ≤ \(\frac{1}{4}\)

⇒ x ≤ \(\frac{1}{4}\)

∴ x ∈ ( -∞,\(\frac{1}{4}\)] ... (1)

Now,

Let us consider the second inequality.

3 – 4x < 0

⇒ 3 – 4x – 3 < 0 – 3

⇒ –4x < –3

⇒ \(\frac{-4x}{4}\) < \(\frac{-3}{4}\)

⇒ -x < \(-\frac{3}{4}\)

⇒ x > \(\frac{3}{4}\)

∴ x ∈ (\(\frac{3}{4}\),∞) ... (2)

From (1) and (2), we get

x ∈ ( -∞,\(\frac{1}{4}\)] ∩ (\(\frac{3}{4}\),∞)

∴ x ∈ ∅

Thus,

There is no solution of the given system of inequations.