the number of integers satisfying the inequality x/x+6<=1/x
Answers
Answer:
Given
log
3
x−1
+
log
3
3
1
2
1
log
3
x
3
+2>0
⟹
log
3
x−1
+
log
3
1−log
3
3
2
3
log
3
x
+2>0
⟹
log
3
x−1
+
0−1
2
3
log
3
x
+2>0
⟹
log
3
x−1
−
2
3
log
3
x+2>0
⟹
log
3
x−1
>
2
3
log
3
x−2
Squaring on both sides
⟹log
3
x−1>(
2
3
log
3
x−2)
2
Let log
3
x=t
⟹t−1>(
2
3
t−2)
2
⟹t−1>
4
9
t
2
+4−2(
2
3
)2t
⟹t−1>
4
9
t
2
+4−6t
⟹
4
9
t
2
+4−6t−t+1<0
⟹
4
9
t
2
−7t+5<0
Roots of the quadratic equation ax
2
+bx+c=0 is given by
2a
−b±
b
2
−4ac
Therefore, roots are
2(
4
9
)
−(−7)±
7
2
−4(
4
9
)5
⟹
(
2
9
)
7±
49−45
⟹
(
2
9
)
7±2
⟹
(
2
9
)
7+2
and
(
2
9
)
7−2
⟹2 and
9
10
are the roots
therefore
4
9
t
2
−7t+5<0⟹(t−2)(t−
9
10
)<0
⟹(t−2)<0 and (t−
9
10
)>0 or
(t−2)>0 and (t−
9
10
)<0 doesn't have a solution
therefore t<2 and t>
9
10
⟹log
3
x<2 and log
3
x>
9
10
⟹x<3
2
and x>3
9
10
⟹x<9 and x>3.39
⟹x∈(3.39,9)
{4,5,6,7,8} are integers which belong to the set (3.39,9)
Therefore the total number of integers =5