Question

Solve each of the following system of inequations in R

(2x-3)/4 - 2 ≥ 4x/3 - 6, 2(2x + 3) < 6(x – 2) + 10

Answer 1

Given,

(2x-3)/4 - 2 ≥ 4x/3 - 6, 

2(2x + 3) < 6(x – 2) + 10

Let us consider the first inequality.

⇒ 3(2x – 11) ≥ 4(4x – 18) 

⇒ 6x – 33 ≥ 16x – 72 

⇒ 6x – 33 + 33 ≥ 16x – 72 + 33 

⇒ 6x ≥ 16x – 39 

⇒ 16x – 39 ≤ 6x 

⇒ 16x – 39 + 39 ≤ 6x + 39 

⇒ 16x ≤ 6x + 39 

⇒ 16x – 6x ≤ 6x + 39 – 6x 

⇒ 10x ≤ 39

⇒ \(\frac{10x}{10}\) ≤ \(\frac{39}{10}\) 

⇒ x ≤ \(\frac{39}{10}\)

∴ x ∈ (-∞,\(\frac{39}{10}\)] ...(1)

Now, 

Let us consider the second inequality. 

2(2x + 3) < 6(x – 2) + 10

⇒ 4x + 6 < 6x – 12 + 10 

⇒ 4x + 6 < 6x – 2 

⇒ 4x + 6 – 6 < 6x – 2 – 6 

⇒ 4x < 6x – 8 

⇒ 6x – 8 > 4x 

⇒ 6x – 8 + 8 > 4x + 8 

⇒ 6x > 4x + 8 

⇒ 6x – 4x > 4x + 8 – 4x 

⇒ 2x > 8

⇒ \(\frac{2x}{2}\) > \(\frac{8}{2}\)

⇒ x > 4 

∴ x ∈ (4, ∞) ...(2) 

From (1) and (2), we get

x ∈ (-∞,\(\frac{39}{10}\)] ∩ (4, ∞)

∴ x ∈ ∅ 

Thus,

There is no solution of the given system of inequations.