Linear Inequations — Maths STD 11 Science — Question
Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLinear Inequations3 Marks
Question
Solve the following system of equations in R.
$|3-4\text{x}|\geq9$
✓
Answer
$|3-4\text{x}|\geq9$
$\Rightarrow4\Big|\frac{3}{4}-\text{x}\Big|\geq9$
$\Rightarrow\Big|\frac{3}{4}-\text{x}\Big|\geq\frac{9}{4}$
Case 1: When $-\infty<\text{x}\leq-\frac{3}{4}$
$\Big|\frac{3}{4}-\text{x}\Big|=\Big(\frac{3}{4}-\text{x}\Big)$
$\therefore\Big|\frac{3}{4}-\text{x}\Big|\geq\frac{9}{4}$
$\Rightarrow\Big(\frac{3}{4}-\text{x}\Big)\geq\frac{9}{4}$
$\Rightarrow-\frac{6}{4}\geq\text{x}$
$\Rightarrow-\frac{3}{2}\geq\text{x}$
But, $-\infty<\text{x}<-1$
$\therefore$ The solution set of the given inequation is $\Big(-\infty,-\frac{3}{2}\Big]$
Case 2: When $-\frac{3}{4}<\text{x}<\infty$
$\Big|\frac{3}{4}-\text{x}\Big|=-\Big(\frac{3}{4}-\text{x}\Big)$
$\therefore\Big|\frac{3}{4}-\text{x}\Big|\geq\frac{9}{4}$
$\Rightarrow\text{x}\geq3$
But, $-\frac{3}{4}<\text{x}<\infty$
$\therefore$ The solution set of the given inequation is $[3,\infty).$
Combining case 1 and case 2,
We obtain that the solution set of given in equality is $\Big(-\infty,-\frac{3}{2}\Big]\cup(3,\infty)$
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