Book: RD Sharma - Mathematics
Chapter: 15. Linear Inequations
Subject: Maths - Class 11th
Q. No. 13 of Exercise 15.2
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13
Solve each of the following system of inequations in R
2(x – 6) < 3x – 7, 11 – 2x < 6 – x
Given 2(x – 6) < 3x – 7 and 11 – 2x < 6 – x
Let us consider the first inequality.
2(x – 6) < 3x – 7
⇒ 2x – 12 < 3x – 7
⇒ 2x – 12 + 12 < 3x – 7 + 12
⇒ 2x < 3x + 5
⇒ 3x + 5 > 2x
⇒ 3x + 5 – 5 > 2x – 5
⇒ 3x > 2x – 5
⇒ 3x – 2x > 2x – 5 – 2x
⇒ x > –5
∴ x ∈ (–5, ∞) (1)
Now, let us consider the second inequality.
11 – 2x < 6 – x
⇒ 11 – 2x – 11 < 6 – x – 11
⇒ –2x < –x – 5
⇒ –x – 5 > –2x
⇒ –x – 5 + 5 > –2x + 5
⇒ –x > –2x + 5
⇒ –x + 2x > –2x + 5 + 2x
⇒ x > 5
∴ x ∈ (5, ∞) (2)
From (1) and (2), we get
x ∈ (–5, ∞) ∩ (5, ∞)
∴ x ∈ (5, ∞)
Thus, the solution of the given system of inequations is (5, ∞).
