Problem 51 - Finite Mathematics and Applied Calculus

Question

There should be at least 3 more grams of ingredient \(\mathrm{A}\) than ingredient \(\mathrm{B}\). (A) \(3 x-y \leq 0\) (B) \(x-3 y \geq 0\) (C) \(x-y \geq 3\) (D) \(3 x-y \geq 0\)

Step-by-Step Solution

Answer

The correct inequality to represent the condition that there should be at least 3 more grams of ingredient A than ingredient B is \(x - y \geq 3\), which corresponds to option (C).

Step 1: Identify the condition mentioned about the amounts of ingredients A and B#

Ingredient A should have at least 3 more grams than ingredient B. This means that the amount of ingredient A (let it be represented by x) minus the amount of ingredient B (let it be represented by y) should be greater than or equal to 3. Mathematically, this relationship can be written as: \( x - y\geq 3\). This is the inequality we are looking for.

Step 2: Compare the inequality obtained with given options#

The inequality we have obtained is: \( x - y\geq 3\) Now we need to compare this with the given options: (A) \(3x - y\leq 0\): This is not the correct inequality as our inequality is in the form of \(x - y\geq 3\). (B) \(x - 3y\geq 0\): In this inequality, the difference bwtween the amounts is multiplied with 3 and hence this is also not correct as it does not represent the required condition. (C) \(x - y\geq 3\): This is the correct inequality as it precisely matches our obtained inequality. (D) \(3x - y\geq 0\): This inequality is also not correct as it does not represent the required condition. Thus, the correct inequality is given by option (C).

Key Concepts

Linear Inequalities

Linear inequalities are similar to linear equations but instead of an equal sign, they use inequality symbols like \(\leq\), \(\geq\), \(<\), or \(>\). These symbols help describe the relationship between two expressions that are not necessarily equal. Linear inequalities have wide applications in problem-solving variables and constraints.

To solve a linear inequality, we follow similar steps as those used for solving a linear equation. However, special care must be taken in the following cases:

• When multiplying or dividing both sides of the inequality by a negative number, the inequality sign flips.
• The solution of the inequality is often a range of values rather than a single number.

For the given problem, we identified the inequality \(x - y \geq 3\). This means the amount of ingredient \(A\) should exceed ingredient \(B\) by at least 3 grams. Note the use of \(\geq 3\) to indicate that \(x\) can be exactly 3 grams more or more than that compared to \(y\).
Understanding how to interpret these types of inequalities helps analyze and solve real-world problems where conditions such as "at least" or "no more than" arise regularly.

Problem Solving in Algebra

Algebra can be used to express relationships mathematically, making it easier to solve complex problems. In our exercise, we formed an inequality to express the condition regarding ingredient \(A\) and \(B\). This is a common strategy used in algebraic problem solving.

Here's a structured approach:

• Understand the Problem: Identify what needs to be solved. In this case, we needed the relationship between two quantities.
• Frame the Equation or Inequality: Translate the words into a mathematical expression or inequality.
• Simplify and Solve: Rearrange terms if necessary to isolate the variable and find the solution.
• Verify: After finding a solution, check if it satisfies all conditions given in the problem.

For the problem at hand, we successfully interpreted the condition "at least 3 more grams" into an inequality. We compared the result to options, confirming the correct option as \(x - y \geq 3\). It's key to understand this process, as recognizing patterns and translating them into mathematical language is central to algebraic reasoning.

Mathematical Modelling

Mathematical modelling is the process of using mathematics to solve real-world problems, which often involves forming equations or inequalities based on a given situation. In this exercise, we took a practical scenario involving ingredients, which was then translated into a mathematical inequality.

When building a model:

• Identify the Variables: Determine what the unknown quantities in the problem are.
• Formulate the Model: Use equations or inequalities to describe the relationships between variables.
• Solve the Model: Use mathematical methods to find solutions to the problem described by the model.
• Interpret the Solution: Relate the mathematical solution back to the real-world context.

For this exercise, \(x\) and \(y\) were our variables for the amounts of ingredients \(A\) and \(B\). The inequality \(x - y \geq 3\) formed our model. This simple yet powerful process demonstrates how algebraic tools can provide insights into and solutions for real-world situations.