Question

Solve each of the following system of inequations in R 

2x + 6 ≥ 0, 4x – 7 < 0

Answer 1

Given,

2x + 6 ≥ 0 and 4x – 7 < 0 

Let us consider the first inequality. 

2x + 6 ≥ 0 

⇒ 2x + 6 – 6 ≥ 0 – 6 

⇒ 2x ≥ –6

⇒ \(\frac{2x}{2}\) ≥ \(\frac{-6}{2}\)

⇒ x ≥ –3 

∴ x ∈ [–3, ∞) ...(1) 

Now,

Let us consider the second inequality. 

4x – 7 < 0 

⇒ 4x – 7 + 7 < 0 + 7 

⇒ 4x < 7

⇒ \(\frac{4x}{4}\) < \(\frac{7}{4}\)

⇒ x < \(\frac{7}{4}\)

∴ x ∈ (- ∞,\(\frac{7}{4}\)) ....(2)

From (1) and (2), we get

x ∈ [- 3,∞) ∩ (- ∞,\(\frac{7}{4}\))

∴ x ∈ [-3,\(\frac{7}{4}\))

Thus, 

The solution of the given system of inequations is  [-3,\(\frac{7}{4}\)).