Ten observations $5,9,14,15,x+1,2x-13,28,30,32,34$ are written in ascending order. The median of the data is $24$. Find the value of $x$.

The total count of data observations given is $n=10$, which is an even number.For an even number of observations sorted in ascending sequence, the median is calculated as the average of the two middle terms, located at positions $\frac{n}{2}$ and $\left(\frac{n}{2}+1\right)$:
$$\text{Middle terms positions}=\frac{10}{2}=5^{\text{th}}\text{ term}\text{and}5+1=6^{\text{th}}\text{ term}$$
From the sorted list:$5^{\text{th}}\text{ term}=x+1$$6^{\text{th}}\text{ term}=2x-13$ Formulating the median equation:
$$\text{Median}=\frac{(x+1)+(2x-13)}{2}=24$$
$$\frac{3x-12}{2}=24$$
$$3x-12=48$$
$$3x=60⟹x=20$$