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Question

Mathematics Question on inequalities

The number of real roots of the equation ∣2−∣1−∣x∣∣∣=1|2-|1-|x|||=1 is

A

11

B

33

C

55

D

66

Answer

55

Explanation

Solution

∣2−∣1−∣x∣∣∣=1⇒2−∣1−∣x∣∣=±1⇒∣1−∣x∣∣=1\left|2-\right|1-\left|x\right|\left||=1 \Rightarrow 2-\left|1-\right|x \right||=\pm1\Rightarrow \left|1-\right|x\left|\right|=1 or 33 If ∣1−x∣x∣∣=1⇒1−∣x∣=±1⇒∣x∣=0|1-x|x||=1 \Rightarrow 1 - |x|=\pm 1 \Rightarrow |x|=0 or 2⇒x=02 \Rightarrow x=0 or ±2\pm2 If ∣1−∣x∣∣=3⇒1−∣x∣=±3⇒∣x∣=−2|1-|x||=3 \Rightarrow 1-|x|=\pm 3 \Rightarrow |x| =-2 or 44 ⇒∣x∣=4⇒x=±4  [∵∣x∣≠−2]\Rightarrow |x|=4 \Rightarrow x=\pm 4 \,\,[\because |x|\ne -2] ∴\therefore Solution set is \left\\{-4, -2, 0, 2, 4\right\\}, hence 5 real roots in all.