Linear Inequations-5

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Solve the following system of inequations:
2(2x + 3) – 10 < 6(x - 2)
(2x – 3)/4 + 6 ≥ 2 + 4x/3

Solution: The given system of inequations is

2(2x + 3) – 10 < 6(x - 2)

(2x – 3)/4 + 6 ≥ 2 + 4x/3

Now, 2(2x + 3) – 10 < 6(x - 2)

=>4x + 6 – 10 < 6(x - 2)

=> 4x – 4 < 6x – 12

=> 4x – 6x < 4 – 12

=> -2x < -8

=> x > 4

=> x ∈ (4, ∞)

=> So the solution set of the first equation is the interval (4, ∞).

and (2x – 3)/4 + 6 ≥ 2 + 4x/3

(2x – 3 + 24)/4 ≥ (6 + 4x)/3

(2x + 21)/4 ≥ (4x + 6)/3

=>3(2x + 21) ≥ 4(4x + 6)

=> 6x + 63 ≥ 16x + 24

=> 6x – 16x ≥ 24 – 63

=> -10x ≥ -39

=> x ≤ 39/10

=> x ≤ 3.9

=> x ∈ (-∞. 3.9]

So, the solution set of the inequation (ii) is (-∞. 3.9]

The solution sets of inequations (i) and (ii) are graphed on real line in the following figures.

⟨(-∞)-------------------(0)--------------------(3.9)-(4)------------------------------------------⟩

⟨(-∞)------------------(0)--------------------(3.9)-(4)----------------------------------------(+∞)⟩

We observe that there is no common solution of the two inequations. SO, the given system of inequations has no solution.

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