Linear Inequations — Maths STD 11 Science — Question
CBSE BoardEnglish MediumSTD 11 ScienceMathsLinear Inequations1 Mark
Question
Write the solution set of the inequation $|\text{x}-1|\geq|\text{x}-3|$
✓
Answer
$|\text{x}-1|\geq|\text{x}-3|$
$\Rightarrow|\text{x}-1|-|\text{x}-3|\geq0$
By equating the expression within the modulus to zero, we get x = 1, 3.
These point divide real line in three parts viz. $(-\infty,1),[1,3)$ and $[3\infty).$
Case 1: When $-\infty<\text{x}<1$
|x - 1| = -(x - 1) and |x - 3| = -(x - 3)
$\therefore|\text{x}-1|-|\text{x}-3|\geq0$
$\Rightarrow-2\geq0$ which is not true.
So, the given inequation has no solution for $\text{x}\in(-\infty,1)$
Case 2: When $1\geq\text{x}<3$
|x - 1| = -(x - 1) and |x - 3| = -(x - 3)
$\therefore|\text{x}-1|-|\text{x}-3|\geq0$
⇒ (x - 1) + (x - 3) $\geq$ 0
$\Rightarrow2\text{x}-4\geq0$
$\Rightarrow\text{x}\geq2$
But $1\leq\text{x}\leq3$
Therefore in case the solution set of the given inequation is [2, 3)
Case 3: When $3\leq\text{x}<\infty$
|x - 1| = -(x - 1) and |x - 3| = -(x - 3)
$\therefore|\text{x}-1|-|\text{x}-3|\geq0$
⇒ (x - 1) + (x - 3) $\geq$ 0
$\Rightarrow2\geq0$
The solution set of the given inequation is $[3,\infty)$
Combining 1 and 3 we obtain that the solution set of the given inequation is $[2,3)\cup[3,\infty)=[2,\infty)$
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