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in Linear Inequations by (23.5k points)
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Find the complete set of values that satisfy the inequalities | |x| – 3 | < 2 and | |x| – 2| < 3.

(a) (– 5, 5) 

(b) (– 5, – 1) ∪ (1, 5) 

(c) (1, 5) 

(d) (– 1, 1)

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(a) (– 5, – 1) ∪ (1, 5)

Let |x| = p, where p > 0            …(i) 

So |p – 3| < 2 and |p – 2| < 3            …(ii) 

⇒ 1 < p < 5 and – 1 < p < 5             …(iii) 

(∴ |x – a| < r ⇒ a – r < x < a + r) 

Therefore, the conditions (i), (ii) and (iii) are satisfied by 1 < p < 5, i.e. 1 < | x | < 5, i.e., | x | > 1 and | x | < 5 

i.e., x < – 1 or x > 1 and – 5 < x < 5 

\(x\)∈ (– 5, – 1) ∪ (1, 5).

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