Solve each of the following system of inequations in R 2x + 5 ≤ 0,

1.	Solve each of the following system of inequations in R 2x + 5 ≤ 0, x – 3 ≤ 0
Answer» Given, 2x + 5 ≤ 0 and x – 3 ≤ 0 Let us consider the first inequality. 2x + 5 ≤ 0 ⇒ 2x + 5 – 5 ≤ 0 – 5 ⇒ 2x ≤ –5 ⇒ \(\frac{2x}{2}\) ≤ \(\frac{-5}{2}\) ⇒ x ≤ \(-\frac{5}{2}\) ∴ x ∈ (- ∞,\(-\frac{5}{2}\)] ...(1) Now, Let us consider the second inequality. x – 3 ≤ 0 ⇒ x – 3 + 3 ≤ 0 + 3 ⇒ x ≤ 3 ∴ x ∈ (–∞, 3] ...(2) From (1) and (2), we get x ∈ (-∞,\(-\frac{5}{2}\)] ∩ (-∞, 3] ∴ x ∈ (-∞,\(-\frac{5}{2}\)] Thus, The solution of the given system of inequations is (-∞,\(-\frac{5}{2}\)].

Solve each of the following system of inequations in R 2x + 5 ≤ 0, x – 3 ≤ 0

Answer»

Given,

2x + 5 ≤ 0 and x – 3 ≤ 0

Let us consider the first inequality.

2x + 5 ≤ 0

⇒ 2x + 5 – 5 ≤ 0 – 5

⇒ 2x ≤ –5

⇒ \(\frac{2x}{2}\) ≤ \(\frac{-5}{2}\)

⇒ x ≤ \(-\frac{5}{2}\)

∴ x ∈ (- ∞,\(-\frac{5}{2}\)] ...(1)

Now,

Let us consider the second inequality.

x – 3 ≤ 0

⇒ x – 3 + 3 ≤ 0 + 3

⇒ x ≤ 3

∴ x ∈ (–∞, 3] ...(2)

From (1) and (2), we get

x ∈ (-∞,\(-\frac{5}{2}\)] ∩ (-∞, 3]

∴ x ∈ (-∞,\(-\frac{5}{2}\)]

Thus,

The solution of the given system of inequations is (-∞,\(-\frac{5}{2}\)].