If |2x - 3| |x + 5|, then x lies in the interval

Question

If $| 2 x - 3 |$ $<$ $| x + 5 |$ , then $x$ lies in the interval

Smart Achievers · Accepted Answer

(3)

$We have,$ $| 2 x - 3 |$ $<$ $| x + 5 |$ ...(i)

$Let x \geq \frac{3}{2}, then 2 x - 3 \geq 0 and x + 5 \geq 0$

$Thus, 2 x - 3 < x + 5 \Rightarrow x < 8$

$Thus x \in [\frac{3}{2}, 8) satisfies the above inequality for x \geq \frac{3}{2}$

$Now, let - 5 \leq x < \frac{3}{2}$

$Then, 2 x - 3 < 0 and x + 5 \geq 0$

$So, 3 - 2 x < x + 5 \Rightarrow 3 x > - 2 \Rightarrow x > - \frac{2}{3}$

$But, - 5 \leq x < \frac{3}{2}$

$Thus, x \in (- \frac{2}{3}, \frac{3}{2}) satisfies the given inequality (i).$

$Also, let x < - 5 . Then, 2 x - 3 < 0 and x + 5 < 0$

$\Rightarrow 3 - 2 x < - 5 - x \Rightarrow x > 8$ ...(ii)

$However, x < - 5$

$∴$ $The above inequality (ii) does not hold.$

$We have checked all the required intervals.$

$∴$ $x \in [\frac{3}{2}, 8) \cup (- \frac{2}{3}, \frac{3}{2}) \Rightarrow x \in (- \frac{2}{3}, 8)$

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