Question

A party planner is reserving rooms for a corporate event and has at most $1,400 to spend on renting the space. The Sunrise Room costs $160 per hour and the Twilight Room costs $230 per hour.
Write the inequality in standard form that describes this situation. Use the given numbers and the following variables.
x = the number of hours the Sunrise Room is reserved
y = the number of hours the Twilight Room is reserved

Answer 1

To represent the spending limit for renting the rooms, we can use the following information:

- The cost for the Sunrise Room is $160 per hour.
- The cost for the Twilight Room is $230 per hour.
- The total budget is $1,400.

Let \( x \) be the number of hours the Sunrise Room is reserved, and \( y \) be the number of hours the Twilight Room is reserved.

The cost for the Sunrise Room for \( x \) hours would be \( 160x \), and the cost for the Twilight Room for \( y \) hours would be \( 230y \).

The total cost for both rooms combined must be less than or equal to the budget of $1,400. Thus, the inequality can be expressed as:

\[
160x + 230y \leq 1400
\]

To convert this inequality into standard form (which typically takes the form \( Ax + By \leq C \)), we can rewrite it as:

\[
160x + 230y \leq 1400
\]

This is already in standard form as required. If needed, we could also rearrange to fit the inequality in the form \( Ax + By + C \leq 0 \):

\[
160x + 230y - 1400 \leq 0
\]

Either format directly captures the spending limits on reserving the rooms. The final standard form inequality is:

\[
160x + 230y \leq 1400
\]