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Sets and Relations — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsSets and Relations2 Marks
Question
Solve the following inequalities and write the solution set using interval notation.$6 x^2+1 \leq 5 x$
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Answer

$6 x^2+1 \leq 5 x$
$6 x^2-5 x+1 \leq 0$
$6 x^2-3 x-2 x+1 \leq 0$
$(3 x-1)(2 x-1) \leq 0$ either 3x – 1 ≤ 0 and 2x – 1 ≥ 0 or 3x – 1 ≥ 0 and 2x – 1 ≤ 0 Case I: 3x – 1 ≤ 0 and 2x – 1 ≥ 0
$\therefore x \leq \frac{1}{3}$ and $x \geq \frac{1}{2}$, which is not possible.
Case II:
$3 x-1 \geq 0 \text { and } 2 x-1 \leq 0$
$\therefore x \geq \frac{1}{3} \text { and } x \leq \frac{1}{2}$
$\therefore x \in\left[\frac{1}{3}, \frac{1}{2}\right]$

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