Solve the inequality
(i)(log 2 x) 2 – |(log 2 x) – 2| ≥ 0
(ii) 2 | log 3 x | + log 3 x ≥ 3
(iii) Find the complete solution set of 
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) Let log 2 x = t
t 2 – |t – 2| ≥ 0
Case- Ι t ≥ 2
t 2 – t + 2 ≥ 0 ⇒ t ∈ R
Hence t ∈ [2, ∞ )
Case- ΙΙ t < 2
t 2 + t – 2 ≥ 0 ⇒ (t + 2)(t – 1) ≥ 0
⇒ t ∈ (– ∞ , –2] ∪ [1, 2)
Hence t ∈ (– ∞ , –2] ∪ [1, 2)
From Case- Ι & Case- ΙΙ t ∈ (– ∞ , –2] ∪ [1, ∞ )
⇒ log 2 x ∈ (– ∞ , –2] ∪ [1, ∞ )
⇒ x ∈ 
(ii) Let log 3 x ≥ 0 ⇒ x ≥ 1
Inequation become log 3 x ≥ 1 ⇒ x ≥ 3
If log 3 x ≤ 0 ⇒ x ∈ [0, 1]
Inequation becomes –log 3 x ≥ 3 ⇒ 0 < x ≤ 
so x ∈ 
log 3 x ≥ 0 ⇒ x ≥ 1
log 3 x ≥ 1 ⇒ x ≥ 3
; log 3 x ≤ 0 ⇒ x ∈ [0, 1]
–log 3 x ≥ 3 ⇒ 0 < x ≤ 
x ∈ 
(iii) Case-I : x ≥ 0 ⇒ 2 x+1 ≥ 2 3/2 ⇒ x ≥ 
Case-II : x ≤ 0 ⇒ 2 x + 2 –x ≥ 2 3/2
Let 2 x = y ⇒ y ⇒ y 2 –2 
y + 1 ≥ 0
⇒ y = 
or 2 x ≥ 
+ 1 (projected as x < 0)
⇒ x ≤ log 2 ( 
)
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