Answer
$-(\dfrac{21}{2})x^6$ or, $-10.5x^6$
Work Step by Step
Our aim is to find the fourth term for $(x-\dfrac{1}{2})^{9}$.
General formula:$(p+q)^n=\displaystyle \binom{n}{k}p^{n-k}q^k$
and $\displaystyle \binom{n}{k}=\dfrac{n!}{k!(n-k)!}$
Now, $(x-\dfrac{1}{2})^{9}=\displaystyle \binom{9}{3}(x)^{9-3}(-\dfrac{1}{2})^3$
This implies,
$=\dfrac{9!}{3!(9-3)!}(x)^{6}(-\dfrac{1}{2})^3$
$=\dfrac{ 9 \times 8 \times 7 \times 6!}{(3 \times 2 \times 1)6!}x^{6}(-\dfrac{1}{8})$
Thus,
$=-(\dfrac{21}{2})x^6$ or, $-10.5x^6$
