Question

Jose and his children went into a grocery store where they sell apples for $2.25 each and mangos for $1.25 each. Jose has $20 to spend and must buy at least 9 apples and mangos altogether. Also, he must buy at least 3 apples and at most 9 mangos. If � x represents the number of apples purchased and � y represents the number of mangos purchased, write and solve a system of inequalities graphically and determine one possible solution.

Answer 1

We defined a system of inequalities for the apples and mangos that Jose can buy with his $20, taking into account the minimum and maximum quantity constraints. After graphing these inequalities, the solution will be any point in the shaded region that also has integer values for x and y.

Step-by-step explanation:

To solve this problem graphically, we need to create a system of inequalities based on the given constraints and then find a region that satisfies all these conditions. Let's define our variables: let × represents the number of apples and × represents the number of mangos. Here are the inequalities representing Jose's shopping conditions:

• 
  
• Jose must buy at least 9 fruits in total: x + y ≥ 9.
• 
  
• Jose must buy at least 3 apples: x ≥ 3.
• 
  
• Jose can buy at most 9 mangos: y ≤ 9.
• 
  
• The cost of the apples and mangos should not exceed $20: 2.25x + 1.25y ≤ 20.

To graph these inequalities, each will be represented by a line on a graph with x (apples) on the horizontal axis and y (mangos) on the vertical axis. The area of the graph that satisfies all inequalities will be shaded and will represent all possible solutions. The vertices of the shaded region are possible solutions to the system but we'll need to find integer solutions since we can't buy fractional parts of a fruit.

Answer 2

Answer: $15.25

Step-by-step explanation:

Let x be the number of apples purchased and y be the number of mangos purchased. Then we have the following constraints:

2.25x + 1.25y ≤ 20 (Jose has $20 to spend)

x + y ≥ 9 (Jose must buy at least 9 apples and mangos altogether)

x ≥ 3 (Jose must buy at least 3 apples)

y ≤ 9 (Jose can buy at most 9 mangos)

To solve this system of inequalities graphically, we can plot the relevant lines and shade the feasible region.

First, we graph the line 2.25x + 1.25y = 20 by finding its intercepts:

When x = 0, we have 1.25y = 20, so y = 16.

When y = 0, we have 2.25x = 20, so x = 8.89 (rounded to two decimal places).

Plotting these intercepts and connecting them with a line, we get:

| *

16 | *

|

| /

| /

|/

0 *--------*

0 8.89

Next, we graph the line x + y = 9 by finding its intercepts:

When x = 0, we have y = 9.

When y = 0, we have x = 9.

Plotting these intercepts and connecting them with a line, we get:

|

9 | *

|

| /

| /

|/

0 *--------*

0 9

Finally, we shade the feasible region by considering the remaining constraints:

x ≥ 3: This means we shade to the right of the line x = 3.

y ≤ 9: This means we shade below the line y = 9.

Shading these regions and finding their intersection, we get:

| *

16 | * |

| |

| / |

| / |

|/ |

9 *--------*---

| | /

| |/

| *

0 *--------*

3 8.89

The feasible region is the shaded triangle bounded by the lines 2.25x + 1.25y = 20, x + y = 9, and x = 3.

To find one possible solution, we can pick any point within the feasible region. For example, the point (4, 5) satisfies all the constraints and represents buying 4 apples and 5 mangos, which costs 4(2.25) + 5(1.25) = $15.25.