Select The Inequality In Standard Form That Describes The Scenario Where Lasper Buys Cupcakes And Brownies With A Budget Of $21.

Navigating the sweet world of bakery treats while staying within budget can be a delicious challenge. In this article, we'll delve into a scenario where Lasper, a generous friend with a $21 limit, aims to buy cupcakes and brownies. Our mission is to translate this real-world situation into a mathematical inequality in standard form. We'll explore the process step-by-step, ensuring you understand how to represent constraints and make informed decisions, especially when delectable treats are involved. So, let's embark on this mathematical journey to help Lasper make the most of his bakery budget!

Understanding the Problem

Before we dive into the mathematical representation, let's break down the scenario. Lasper's budget is our primary constraint: he can spend no more than $21. This means his total spending must be less than or equal to $21. The bakery offers two tempting options: cupcakes at $1 each and brownies at $2 each. Lasper wants to buy a combination of these treats to share with his friends. The key is to find an inequality that captures all these conditions.

To clearly define the problem, we need to identify the variables. Let's use 'x' to represent the number of cupcakes Lasper buys and 'y' to represent the number of brownies. With these variables in place, we can start building our inequality. The cost of cupcakes will be 1 * x (or simply x), and the cost of brownies will be 2 * y (or 2y). The total cost will be the sum of these two, and this sum must be less than or equal to $21.

This careful setup is crucial in translating word problems into mathematical expressions. By understanding the constraints and defining the variables, we lay the groundwork for a successful solution. In the next section, we will construct the inequality that represents Lasper's budgetary constraints, ensuring he can treat his friends without breaking the bank.

Crafting the Inequality in Standard Form

Now that we have a clear understanding of the problem and our variables defined, let's translate the scenario into a mathematical inequality. Remember, Lasper's total spending on cupcakes and brownies must be less than or equal to $21. We know that each cupcake costs $1 (represented as x) and each brownie costs $2 (represented as 2y). Therefore, the total cost can be expressed as x + 2y.

To incorporate Lasper's budget constraint, we set the total cost less than or equal to $21. This gives us the inequality: x + 2y ≤ 21. This inequality represents all the possible combinations of cupcakes (x) and brownies (y) that Lasper can buy without exceeding his $21 budget.

However, the question asks for the inequality in standard form. The standard form for a linear inequality is Ax + By ≤ C, where A, B, and C are constants. Our current inequality, x + 2y ≤ 21, is already in this form! Here, A = 1, B = 2, and C = 21. This means we've successfully translated Lasper's bakery dilemma into a standard form inequality.

This inequality is a powerful tool. It allows us to visualize and understand the possible solutions. For example, if Lasper buys 5 cupcakes (x = 5), we can substitute this value into the inequality and solve for y to find the maximum number of brownies he can afford. In the next section, we'll explore how to interpret and use this inequality to make informed decisions.

Interpreting the Inequality and Possible Solutions

Our inequality, x + 2y ≤ 21, is more than just a mathematical expression; it's a guide to Lasper's bakery options. This inequality defines a region of possible solutions, each representing a combination of cupcakes (x) and brownies (y) that Lasper can purchase within his budget. To understand this better, let's consider some examples.

Suppose Lasper decides to buy only cupcakes. This means y = 0. Substituting this into our inequality, we get x + 2(0) ≤ 21, which simplifies to x ≤ 21. So, Lasper could buy up to 21 cupcakes if he buys no brownies.

Now, let's say Lasper wants to buy only brownies. This means x = 0. Substituting this into our inequality, we get 0 + 2y ≤ 21, which simplifies to 2y ≤ 21. Dividing both sides by 2, we get y ≤ 10.5. Since Lasper can't buy half a brownie, he can buy a maximum of 10 brownies if he buys no cupcakes.

But what about combinations? Let's say Lasper wants to buy 5 cupcakes (x = 5). Substituting this into our inequality, we get 5 + 2y ≤ 21. Subtracting 5 from both sides, we get 2y ≤ 16. Dividing both sides by 2, we get y ≤ 8. This means Lasper can buy up to 8 brownies if he buys 5 cupcakes.

These examples illustrate how the inequality helps us find various combinations of cupcakes and brownies within Lasper's budget. Each point (x, y) that satisfies the inequality is a feasible solution. In the next section, we'll discuss how to represent these solutions graphically, providing a visual understanding of Lasper's options.

Visualizing the Solution Set

To gain a more comprehensive understanding of Lasper's bakery options, let's visualize the solution set of our inequality, x + 2y ≤ 21. Graphing this inequality will provide a visual representation of all possible combinations of cupcakes and brownies that Lasper can afford.

First, we treat the inequality as an equation: x + 2y = 21. This represents a line on a graph. To plot this line, we can find two points that satisfy the equation. We already found two such points in the previous section: (21, 0) – where Lasper buys 21 cupcakes and 0 brownies – and (0, 10.5) – where Lasper buys 0 cupcakes and 10.5 brownies. Since we can't buy half a brownie, we'll use (0, 10) instead.

Plot these two points on a graph with the x-axis representing the number of cupcakes and the y-axis representing the number of brownies. Draw a line through these points. This line represents the boundary of our solution set. Since our inequality is x + 2y ≤ 21, we are interested in the region below the line (including the line itself), as these points satisfy the 'less than or equal to' condition.

However, we also have the constraints that Lasper cannot buy a negative number of cupcakes or brownies. This means x ≥ 0 and y ≥ 0. These inequalities restrict our solution set to the first quadrant of the graph. The feasible region is the area bounded by the line x + 2y = 21, the x-axis, and the y-axis. Every point within this region represents a valid combination of cupcakes and brownies that Lasper can buy within his $21 budget.

This graphical representation provides a clear picture of Lasper's choices. He can pick any combination of cupcakes and brownies within the shaded region and stay within his budget. In the final section, we'll summarize our findings and discuss the practical implications of this inequality for Lasper's bakery trip.

Conclusion: Lasper's Sweet Success

In this article, we've successfully translated Lasper's bakery budget dilemma into a mathematical inequality and explored its implications. We started with a real-world scenario, identified the key constraints, defined variables, and crafted the inequality x + 2y ≤ 21 in standard form. This inequality represents all possible combinations of cupcakes (x) and brownies (y) that Lasper can purchase without exceeding his $21 budget.

We then delved into interpreting the inequality, understanding that it defines a region of possible solutions. We explored specific scenarios, such as Lasper buying only cupcakes or only brownies, and calculated the maximum quantities he could afford. We also considered mixed combinations, illustrating how the inequality guides Lasper's choices.

Finally, we visualized the solution set by graphing the inequality. This graphical representation provided a clear picture of Lasper's options, showing the feasible region where all valid combinations of cupcakes and brownies lie. This visual aid helps Lasper make informed decisions based on his preferences and budget.

By understanding and applying this inequality, Lasper can confidently navigate the bakery, select a delightful assortment of treats for his friends, and stay within his budget. This exercise demonstrates the power of mathematics in everyday decision-making, turning a simple bakery trip into a sweet success.

In summary, the inequality x + 2y ≤ 21 is the key to Lasper's bakery adventure. It's a testament to how mathematical tools can help us make informed choices and solve real-world problems, one cupcake and brownie at a time.