#### Answer

No, the person who is $5$ feet $8$ inches tall and weighs $135$ pounds is not in the healthy weight region.

#### Work Step by Step

It is known that, $1\text{ feet}=\text{12 inches}$
Let us consider height in feet and convert it into inches as shown below:
$\begin{align}
& \text{5 feet 8 inches}=\text{5}\times \text{12 inches}+\text{8inches} \\
& =\text{68 inches}
\end{align}$
The coordinates which define the height and weight of a person are $\left( 68,135 \right)$. So, in order to check whether this point lies in the healthy weight region or not, substitute the coordinates for the x and y variables respectively in both the provided equations as shown below:
Put the values in the first equation as given below:
$\begin{align}
& 5.3x-y\ge 180 \\
& 5.3\left( 68 \right)-135\ge 180 \\
& 225.4\ge 180
\end{align}$
And the inequality holds.
Now, put the values in the second equation as given below:
$\begin{align}
& 4.1\left( x \right)-y\le 140 \\
& 4.1\left( 68 \right)-135\le 140 \\
& 143.8\le 140
\end{align}$
Which is incorrect. Thus, the inequality does not hold.
Therefore, both equations are not satisfied by the coordinates of a person's height and weight, which means that the provided point does not lie in the healthy weight region.
Hence, a person who is $5$ feet $8$ inches tall and weighs $135$ pounds is not in the healthy weight region.