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Find the solution set for x ∈ R which satisfies both the inequation: 2(3x - 4) - 2 < 4x - 2 ≥ 2x – 4 and 5x – 3 < 3x + 1.

Find the solution set for x ∈ R which satisfies both the inequation: 2(3x - 4) - 2 < 4x - 2 ≥ 2x – 4 and 5x – 3 < 3x + 1. Correct Answer [-1, 2)

Concept:

These are some of the rules, which needs to be remembered while solving in equations:

  • When equal numbers are added (or subtracted) on (from) both the sides of inequalities then the sign of inequality does not changes.

 

  • When both the sides of the inequality is multiplies or divided by the same positive number then the sign of inequality does not changes.

 

  • Whereas, if both the sides are multiplied or divided by the same negative number then the sign of inequality gets reversed.

Calculation:

Given: x ∈ R satisfies both the in equations: 2(3x - 4) - 2 < 4x - 2 ≥ 2x – 4 and 5x – 3 < 3x + 1

Let’s solve:  2(3x - 4) - 2 < 4x - 2 ≥ 2x – 4

2(3x - 4) -2 < 4x – 2

⇒ 6x - 10 < 4x - 2

⇒ 2x < 8

⇒ x < 4 ---------------(1)

4x - 2 ≥ 2x - 4

⇒ 2x ≥ - 2

⇒ x ≥ - 1 -------------(2)

So, from (1) and (2), we get:  -1 ≤ x < 4 i.e x ∈ [-1, 4) --------(3)

Now, let’s solve: 5x – 3 < 3x + 1

5x – 3 < 3x + 1

⇒ (5x - 3) – 3x < (3x + 1) – 3x

⇒ 2x – 3 < 1

Now by adding 3 on both the sides of the above inequality we get

⇒ 2x < 4

⇒ x < 2

⇒ x ∈ (- ∞, 2)----------(4)

The graphical representation of (3) and (4)

[ alt="F2 A.K 26.5.20 Pallavi D1" src="//storage.googleapis.com/tb-img/production/20/05/F2_A.K_26.5.20_Pallavi_D1.png" style="width: 287px; height: 40px;">

So, from (3) and (4) we can conclude that the solution set = [-1, 2)

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