Solve each of the following system of inequations in R 2x + 6 ≥ 0,

1.	Solve each of the following system of inequations in R 2x + 6 ≥ 0, 4x – 7 < 0
Answer» 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,\(\frac{7}{4}\)).

Solve each of the following system of inequations in R 2x + 6 ≥ 0, 4x – 7 < 0

Answer»

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,\(\frac{7}{4}\)).