Question

Solve each of the following system of inequations in R.

5x – 7 < 3(x + 3), 1 - 3x/2 ≥ x - 4

Answer 1

Given,

5x – 7 < 3(x + 3) and 1 - 3x/2 ≥ x - 4

Let us consider the first inequality. 

5x – 7 < 3(x + 3) 

⇒ 5x – 7 < 3x + 9 

⇒ 5x – 7 + 7 < 3x + 9 + 7 

⇒ 5x < 3x + 16 

⇒ 5x – 3x < 3x + 16 – 3x 

⇒ 2x < 16

⇒ \(\frac{2x}{2}\) < \(\frac{16}{2}\) 

⇒ x < 8 

∴ x ∈ (–∞, 8) ...(1) 

Now,

Let us consider the second inequality.

1 - \(\frac{3x}{2}\) ≥ x - 4

⇒ \(\frac{2-3x}{2}\) ≥ x - 4

⇒ \((\frac{2-3x}{2})\) \(\times\) 2 ≥ (x - 4) \(\times\) 2

⇒ 2 – 3x ≥ 2(x – 4) 

⇒ 2 – 3x ≥ 2x – 8 

⇒ 2 – 3x – 2 ≥ 2x – 8 – 2 

⇒ –3x ≥ 2x – 10 

⇒ 2x – 10 ≤ –3x 

⇒ 2x – 10 + 10 ≤ –3x + 10 

⇒ 2x ≤ –3x + 10 

⇒ 2x + 3x ≤ –6x + 10 + 6x 

⇒ 5x ≤ 10

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

⇒ x ≤ 2 

∴ x ∈ (–∞, 2] ....(2) 

From (1) and (2), we get 

x ∈ (–∞, 8) ∩ (–∞, 2] 

∴ x ∈ (–∞, 2] 

Thus,

The solution of the given system of inequations is (–∞, 2].