The set of values of a for which the inequality (x−3a)(x−a−3) < 0 is satisfied for all x in the interval 1 ≤ x ≤ 3 are ____.

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Detailed Solution

The given inequality (x−3a)(x−a−3) < 0 satisfies for all x in interval 1 ≤ x ≤ 3
Hence, x = 1, 2, 3 should also satisfy the given inequality. We find solutions for x = 1,2,3
x = 1 gives (1−3a)(−a−2) < 0 or, (1−3a)(a+2) > 0
Solution for above inequality are a > −2 and a <

\frac{1}{3}

………(i)
x = 2 or, (2−3a)(−a−1) < 0
or, (2−3a)(a+1) > 0
or, Solution for above inequality are a > −1 and a <

\frac{2}{3}

………(ii)
Again,
x=3 or, (3−3a)(−a) < 0 or, (3−3a)(a) > 0
Solution for above inequality are a > 0 and a < 1…….(iii)
(i), (ii) and (iii) equations give solutions to be a∈(0,

\frac{1}{3}

) and a∈(−2, 0).

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