Larry has a budget of [tex]\$220[/tex] to rent a party space. The function [tex]F(h) = 20h + 100[/tex] models the cost of renting the space for [tex]h[/tex] hours.

Which inequality shows all the values of [tex]h[/tex] for which the cost of the rental will be within the budget of [tex]\$220[/tex]?

A. [tex]0 < h \leq 5[/tex]
B. [tex]0 < h \leq 6[/tex]
C. [tex]0 < h \leq 11[/tex]
D. [tex]0 < h \leq 20[/tex]

We are given that the cost to rent the party space for [tex]$h$[/tex] hours is modeled by the function
[tex]$$
F(h) = 20h + 100
$$[/tex]
and Larry has a budget of [tex]$\$[/tex]220[tex]$. This means the total cost must satisfy the inequality
$[/tex][tex]$
20h + 100 \leq 220.
$[/tex][tex]$

Step 1: Set Up the Inequality

Since the rental cost must not exceed Larry's budget, we have:
$[/tex][tex]$
20h + 100 \leq 220.
$[/tex][tex]$

Step 2: Isolate the Term with $[/tex]h[tex]$

Subtract $[/tex]100[tex]$ from both sides of the inequality to isolate the term involving $[/tex]h[tex]$:
$[/tex][tex]$
20h + 100 - 100 \leq 220 - 100,
$[/tex][tex]$
which simplifies to:
$[/tex][tex]$
20h \leq 120.
$[/tex][tex]$

Step 3: Solve for $[/tex]h[tex]$

Divide both sides of the inequality by $[/tex]20[tex]$:
$[/tex][tex]$
\frac{20h}{20} \leq \frac{120}{20},
$[/tex][tex]$
resulting in:
$[/tex][tex]$
h \leq 6.
$[/tex][tex]$

Step 4: Consider the Domain of $[/tex]h[tex]$

Since $[/tex]h[tex]$ represents the number of hours, it must be positive. Therefore, we include the condition:
$[/tex][tex]$
h > 0.
$[/tex][tex]$

Final Inequality

Combining the conditions from Steps 3 and 4, we have:
$[/tex][tex]$
0 < h \leq 6.
$[/tex][tex]$

This inequality represents all the values of $[/tex]h[tex]$ (in hours) for which the cost of the rental will be within Larry's budget of $[/tex]\[tex]$220$[/tex]. Thus, the correct answer is:

[tex]$\boxed{0 < h \leq 6}$[/tex].