Answer :
Certainly! Let's break this problem down step by step.
### Step 1: Define the Inequalities
First, we need to set up the inequalities based on the conditions given:
1. You need to buy at least 5 tickets:
[tex]\[ x + y \geq 5 \][/tex]
2. The total cost should be less than \[tex]$610. Tickets for Atlanta cost \$[/tex]30 each, and tickets for D.C. cost \$50 each:
[tex]\[ 30x + 50y < 610 \][/tex]
### Step 2: Identify Feasible Solutions
These two inequalities represent constraints on the number of tickets you can buy for Atlanta (x) and D.C. (y).
### Step 3: Graph the Inequalities
To graph this system of inequalities, follow these steps:
1. Graph the boundary lines:
- For [tex]\( x + y = 5 \)[/tex]:
[tex]\[ y = 5 - x \][/tex]
- For [tex]\( 30x + 50y = 610 \)[/tex]:
[tex]\[ y = \frac{610 - 30x}{50} \][/tex]
or simplified:
[tex]\[ y = 12.2 - 0.6x \][/tex]
2. Draw the lines:
- The line [tex]\( y = 5 - x \)[/tex] will be a straight line with a negative slope passing through points (0, 5) and (5, 0).
- The line [tex]\( y = 12.2 - 0.6x \)[/tex] will be a straight line with a negative slope passing through points (0, 12.2) and (20.33, 0).
3. Shade the feasible regions:
- For [tex]\( x + y \geq 5 \)[/tex], shade the region above the line [tex]\( y = 5 - x \)[/tex].
- For [tex]\( 30x + 50y < 610 \)[/tex], shade the region below the line [tex]\( y = 12.2 - 0.6x \)[/tex].
4. Identify the intersection of the shaded regions that represents the solution to the system of inequalities.
### Step 4: Analyze Graphical Representation
Let’s plot these to visually identify the solution region.
#### Graphing:
- Equation 1 ([tex]\( y = 5 - x \)[/tex]):
- Point 1: [tex]\( (0, 5) \)[/tex]
- Point 2: [tex]\( (5, 0) \)[/tex]
- Equation 2 ([tex]\( y = 12.2 - 0.6x \)[/tex]):
- Point 1: [tex]\( (0, 12.2) \)[/tex]
- Point 2: [tex]\( (20.33, 0) \)[/tex]
To draw these lines:
1. Draw a straight line from (0, 5) to (5, 0) for [tex]\( y = 5 - x \)[/tex].
2. Draw a straight line from (0, 12.2) to (20.33, 0) for [tex]\( y = 12.2 - 0.6x \)[/tex].
Then, shade the appropriate regions according to the inequalities.
### Step 5: Graphical Solution and Analysis:
You will notice that the feasible region, where both inequalities are satisfied, is the area where the shaded regions overlap. The region of overlap will be above the line [tex]\( y = 5 - x \)[/tex] (inclusive of the boundary because of the "≥" sign) and below the line [tex]\( y = 12.2 - 0.6x \)[/tex] (exclusive of the boundary because of the "<" sign).
### Final Note:
On the graph, it is important to check integer solutions within the shaded region as the number of tickets (x and y) should be whole numbers. This ensures practical ticketing solutions while adhering to the given constraints.
You can calculate or visually identify valid integer points for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that:
[tex]\[ x + y \geq 5 \][/tex]
[tex]\[ 30x + 50y < 610 \][/tex]
Example of solution pairs (x, y) might be (4, 3), (3, 5), etc., based on the graph. Each can then be validated to meet the problem constraints fully.
### Step 1: Define the Inequalities
First, we need to set up the inequalities based on the conditions given:
1. You need to buy at least 5 tickets:
[tex]\[ x + y \geq 5 \][/tex]
2. The total cost should be less than \[tex]$610. Tickets for Atlanta cost \$[/tex]30 each, and tickets for D.C. cost \$50 each:
[tex]\[ 30x + 50y < 610 \][/tex]
### Step 2: Identify Feasible Solutions
These two inequalities represent constraints on the number of tickets you can buy for Atlanta (x) and D.C. (y).
### Step 3: Graph the Inequalities
To graph this system of inequalities, follow these steps:
1. Graph the boundary lines:
- For [tex]\( x + y = 5 \)[/tex]:
[tex]\[ y = 5 - x \][/tex]
- For [tex]\( 30x + 50y = 610 \)[/tex]:
[tex]\[ y = \frac{610 - 30x}{50} \][/tex]
or simplified:
[tex]\[ y = 12.2 - 0.6x \][/tex]
2. Draw the lines:
- The line [tex]\( y = 5 - x \)[/tex] will be a straight line with a negative slope passing through points (0, 5) and (5, 0).
- The line [tex]\( y = 12.2 - 0.6x \)[/tex] will be a straight line with a negative slope passing through points (0, 12.2) and (20.33, 0).
3. Shade the feasible regions:
- For [tex]\( x + y \geq 5 \)[/tex], shade the region above the line [tex]\( y = 5 - x \)[/tex].
- For [tex]\( 30x + 50y < 610 \)[/tex], shade the region below the line [tex]\( y = 12.2 - 0.6x \)[/tex].
4. Identify the intersection of the shaded regions that represents the solution to the system of inequalities.
### Step 4: Analyze Graphical Representation
Let’s plot these to visually identify the solution region.
#### Graphing:
- Equation 1 ([tex]\( y = 5 - x \)[/tex]):
- Point 1: [tex]\( (0, 5) \)[/tex]
- Point 2: [tex]\( (5, 0) \)[/tex]
- Equation 2 ([tex]\( y = 12.2 - 0.6x \)[/tex]):
- Point 1: [tex]\( (0, 12.2) \)[/tex]
- Point 2: [tex]\( (20.33, 0) \)[/tex]
To draw these lines:
1. Draw a straight line from (0, 5) to (5, 0) for [tex]\( y = 5 - x \)[/tex].
2. Draw a straight line from (0, 12.2) to (20.33, 0) for [tex]\( y = 12.2 - 0.6x \)[/tex].
Then, shade the appropriate regions according to the inequalities.
### Step 5: Graphical Solution and Analysis:
You will notice that the feasible region, where both inequalities are satisfied, is the area where the shaded regions overlap. The region of overlap will be above the line [tex]\( y = 5 - x \)[/tex] (inclusive of the boundary because of the "≥" sign) and below the line [tex]\( y = 12.2 - 0.6x \)[/tex] (exclusive of the boundary because of the "<" sign).
### Final Note:
On the graph, it is important to check integer solutions within the shaded region as the number of tickets (x and y) should be whole numbers. This ensures practical ticketing solutions while adhering to the given constraints.
You can calculate or visually identify valid integer points for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that:
[tex]\[ x + y \geq 5 \][/tex]
[tex]\[ 30x + 50y < 610 \][/tex]
Example of solution pairs (x, y) might be (4, 3), (3, 5), etc., based on the graph. Each can then be validated to meet the problem constraints fully.
