Problem: If x is a number satisfying the equation 3x+9−3x−9=3, then x2 is between:
Answer Choices:
A. 55 and 65
B. 65 and 75
C. 75 and 85
D. 85 and 95
E. 95 and 105
Solution:
3x+9−3x−9=3. Cube both sides and obtain
x+9−3(x+9)2/3(x−9)2/3+3(x+9)1/3(x−9)2/3−x+9=27
Simplify to 9=−3(x+9)1/3(x−9)1/3[(x+9)1/3−(x−9)1/3]=−3(x+9)2/3(x−9)1/3⋅3
∴(x2−81)1/3=−1,x2=80.
or
Let f(x)=3x+9−3x−9. We want to find x, so that f(x1)=3. Since f(9)=318<3 and f(8)=317+1>3, then 8<x1<9. We refine our guess by trying next x1=887; f(887)=31787+21>3.∴887<x1<9∴(887)2<x12<81,78<x12<81.