Solve each of the following system of inequations in R0 < \(\frac{-x}{

1.	Solve each of the following system of inequations in R0 < \(\frac{-x}{2}\) < 30 < -x/2 < 3
Answer» Given, 0 < -x/2 < 3 The above inequality can be split into two inequalities. 0 <\(-\frac{x}{2}\) and \(-\frac{x}{2}\) < 3 Let us consider the first inequality. 0 <\(-\frac{x}{2}\) ⇒ 0\(\times\)2 < \(-\frac{x}{2}\)\(\times\)2 ⇒ 0 < –x ⇒ –x > 0 ⇒ x < 0 ∴ x ∈ (–∞, 0) ....(1) Now, Let us consider the second inequality. \(-\frac{x}{2}\) < 3 ⇒ \(-\frac{x}{2}\) \(\times\) 2 < 3 \(\times\)2 ⇒ –x < 6 ⇒ x > –6 ∴ x ∈ (–6, ∞) ...(2) From (1) and (2), we get x ∈ (–∞, 0) ∩ (–6, ∞) ∴ x ∈ (–6, 0) Thus, The solution of the given system of inequations is (–6, 0).

Solve each of the following system of inequations in R0 < \(\frac{-x}{2}\) < 30 < -x/2 < 3

Answer»

Given,

0 < -x/2 < 3

The above inequality can be split into two inequalities.

0 <\(-\frac{x}{2}\) and \(-\frac{x}{2}\) < 3

Let us consider the first inequality.

0 <\(-\frac{x}{2}\)

⇒ 0\(\times\)2 < \(-\frac{x}{2}\)\(\times\)2

⇒ 0 < –x

⇒ –x > 0

⇒ x < 0

∴ x ∈ (–∞, 0) ....(1)

Now,

Let us consider the second inequality.

\(-\frac{x}{2}\) < 3

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

⇒ –x < 6

⇒ x > –6

∴ x ∈ (–6, ∞) ...(2)

From (1) and (2), we get

x ∈ (–∞, 0) ∩ (–6, ∞)

∴ x ∈ (–6, 0)

Thus,

The solution of the given system of inequations is (–6, 0).