The lower bound of $4.6$ rounded to one decimal place is $4.55$.

When we refer to the lower bound of a number rounded to one decimal place, we mean the smallest value that will still round up to that number. In the case of $4.6$, we need to identify the smallest number that, when rounded to one decimal place, results in $4.6$.

To clarify this concept, let’s break it down further. When rounding to one decimal place, we focus on the second decimal place. If the second decimal place is $5$ or higher, we increase the first decimal place by one. Conversely, if the second decimal place is $4$ or lower, we keep the first decimal place unchanged.

For the number $4.6$, the smallest value that will round up to $4.6$ is $4.55$. This is because any number from $4.55$ up to, but not including, $4.65$ will round to $4.6$ when rounded to one decimal place. For instance, the numbers $4.55$, $4.56$, $4.57$, $4.58$, and $4.59$ all round to $4.6$.

Therefore, the lower bound of $4.6$ rounded to one decimal place is $4.55$. This understanding of bounds is valuable in various mathematical and real-world applications, particularly when you need to ensure that measurements fall within a specified range. Grasping the concept of bounds aids in making precise calculations and minimizes errors in estimations.