https://file.aiquickdraw.com/mathbot/file/1730723486943z1a4j0yo.jpg

Let's go through each part of the problem described in the photo. I'll help clarify the requirements and work through each step.

### Problem Breakdown
The initial problem states:
- **Initial charge** for renting the venue: **$679**.
- **Hourly rate** added to the bill: **$49** per hour.
- **Budget** available: **$1,050**.

The task is divided into parts (a) through (d).

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### Solution Outline

#### Part (a)
*Represent the linear relationship using a table, graph, and equation.*

1. **Equation Form**:
   Since we have an initial cost and a rate per hour, this is a linear equation in the form:
   \[
   y = mx + b
   \]
   where:
   - $ x $ is the number of hours the venue is rented.
   - $ y $ is the total cost.
   - $ m = 49 $ (hourly rate).
   - $ b = 679 $ (initial cost).

   So, the equation becomes:
   \[
   y = 49x + 679
   \]

2. **Table**:
   Create a table for various values of $ x $ (hours). Here are some sample values:

   | Hours (x) | Total Cost (y) |
   |-----------|----------------|
   | 0         | 679            |
   | 1         | 728            |
   | 2         | 777            |
   | 3         | 826            |
   | 4         | 875            |

3. **Graph**:
   Plot this table on a graph, with $ x $ (hours) on the horizontal axis and $ y $ (total cost) on the vertical axis. The line should start at $ y = 679 $ when $ x = 0 $ and increase with a slope of 49.

#### Part (b)
*Determine the cost of renting the venue from 3:00 p.m. to 1:30 a.m.*

- This period is **10.5 hours** (from 3:00 p.m. to 1:30 a.m.).
- Substitute $ x = 10.5 $ into the equation $ y = 49x + 679 $:
  \[
  y = 49(10.5) + 679
  \]
  Calculate:
  \[
  y = 514.5 + 679 = 1193.5
  \]
  So, the cost is **$1,193.50**.

#### Part (c)
*Write an inequality to ensure Suzanne doesn’t exceed her budget of $1,050.*

- We want to find the maximum number of hours she can rent without going over $1,050.
- Set up the inequality:
  \[
  49x + 679 \leq 1050
  \]
- Solve for $ x $:
  \[
  49x \leq 1050 - 679
  \]
  \[
  49x \leq 371
  \]
  \[
  x \leq \frac{371}{49}
  \]
  \[
  x \leq 7.57
  \]
  So, she can rent the venue for up to **7 hours** without exceeding her budget.

#### Part (d)
*Solve the inequality and represent the solution on a number line.*

- From part (c), we found that $ x \leq 7.57 $.
- On a number line, this would mean marking from 0 to 7.57, with a closed dot at 7.57 (indicating it’s included in the allowed time).

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Would you like me to go deeper into any of these steps or explain further?

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### Related Questions

1. What would the cost be if she rented the venue for 12 hours?
2. How much extra would she need if she wanted to rent the venue for 10 hours?
3. How does the hourly rate affect the total cost over time?
4. If the initial charge increased, how would that impact her budget constraints?
5. How would the graph look if the hourly rate changed?

### Tip
To understand inequalities better, try visualizing them on a number line, as this helps in seeing the range of possible solutions.

Analyze the problem situation for renting a party venue with an initial charge and an hourly rate, while considering a budget constraint. Solve for parts including setting up a linear equation, calculating costs, creating an inequality, and representing the solution on a number line.

Explore how to solve a real-life problem of budgeting for a party venue rental using linear equations and inequalities. With a fixed initial cost and hourly rate, learn to set up a linear equation, calculate the total cost, and determine the maximum rental time within a budget. Suitable for students in Grades 8-10.

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