Logarithms

logx+‌

7

2

(‌

x−7

2x−3

)2≥0
Feasible region: x+‌

7

2

>0⇒x>−‌

7

2

And x+‌

7

2

≠1⇒x≠‌

−5

2

Taking intersection: x∈(‌

−7

2

,∞)−{−‌

5

2

,‌

3

2

,7}
Now logab≥0 if a>1 and b≥1
a∈(0,1) and b∈(0,1)
C−I;x+‌

7

2

>1 and (‌

x−7

2x−3

)2≥1
‌x>−‌

5

2

;(2x−3)2−(x−7)2≤0
‌(2x−3+x−7)(2x−3−x+7)≤0
‌(3x−10)(x+4)≤0
‌x∈[−4,‌

10

3

]
Intersection: x∈(‌

−5

2

,‌

10

3

]
‌C−Π:x+‌

7

2

∈(0,1)‌ and ‌(‌

x−7

2x−3

)2∈(0,1)
‌0<x+‌

7

2

<1;(‌

x−7

2x−3

)2<1
‌−‌

7

2

<x<‌

−5

2

;(x−7)2<(2x−3)2
‌x∈(−∞,−4)∪(‌

10

3

,∞)
No common values of x.
Hence intersection with feasible region
We get x∈(‌

−5

2

,‌

10

3

]−{‌

3

2

}
Integral value of x are {−2,−1,0,1,2,3}
No. of integral values =6