Solving Inequalities And Word Problems A Mathematical Exploration

28. Solving the Inequality: rac{1}{2}x + 3 > rac{3}{2}

When we delve into the realm of inequalities, our primary goal is to determine the range of values that satisfy a given expression rather than pinpointing a single solution, as we do with equations. In this specific instance, we are presented with the inequality rac{1}{2}x + 3 > rac{3}{2}, and our mission is to identify all values of x within the domain of rational numbers (Q) that make this statement true. To accomplish this, we'll employ a series of algebraic manipulations, mirroring the techniques used to solve equations, with a crucial consideration for how operations affect the inequality sign.

Our initial step involves isolating the term containing x on one side of the inequality. This is achieved by subtracting 3 from both sides of the expression. This maintains the balance of the inequality, ensuring that the relationship between the two sides remains intact. Subtracting 3 from both sides yields: rac{1}{2}x + 3 - 3 > rac{3}{2} - 3. Simplifying this, we get rac{1}{2}x > rac{3}{2} - rac{6}{2}, which further simplifies to rac{1}{2}x > -rac{3}{2}. This transformation brings us closer to isolating x and understanding its possible values.

Now that we have rac{1}{2}x > -rac{3}{2}, the next logical step is to eliminate the fraction multiplying x. To do this, we multiply both sides of the inequality by 2. Multiplication, in this context, is a straightforward operation that preserves the inequality's direction because we are multiplying by a positive number. If we were to multiply by a negative number, we would need to flip the inequality sign, a critical rule in inequality manipulations. Multiplying both sides by 2 gives us 2 * (rac{1}{2}x) > 2 * (-rac{3}{2}), which simplifies to x > -3. This resulting inequality provides the solution set for our original problem.

The solution x > -3 signifies that any rational number x that is greater than -3 will satisfy the original inequality. This is an infinite set of values, demonstrating a fundamental difference between inequalities and equations. While an equation often has a discrete set of solutions, an inequality defines a continuous range. Considering the options provided, we can see that option D, x > -3, aligns perfectly with our derived solution. Therefore, the correct answer to the inequality rac{1}{2}x + 3 > rac{3}{2} in the domain of Q is indeed x > -3.

In conclusion, solving inequalities requires a solid understanding of algebraic manipulations and how these operations affect the inequality sign. Subtracting or adding the same value from both sides preserves the sign, while multiplying or dividing by a positive number also maintains the sign. However, multiplying or dividing by a negative number necessitates flipping the inequality sign to maintain the truth of the statement. This careful approach ensures that we arrive at the correct solution set, which in this case is all rational numbers greater than -3.

29. Solving a Word Problem: Apples and Mangoes

Word problems form a crucial part of mathematics education as they bridge the gap between abstract mathematical concepts and real-world scenarios. This particular problem involves a scenario in a shop where there are 540 apples and mangoes in total. The problem introduces a relationship between the number of apples and mangoes: “Two times the number of apples is 120 more than the number of mangoes.” Our task is to translate this verbal information into mathematical equations and then solve these equations to find the number of each type of fruit. This process involves algebraic thinking, problem-solving strategies, and a clear understanding of how to represent quantities and relationships using variables and equations.

The initial step in tackling this word problem is to define our variables. Let's denote the number of apples as A and the number of mangoes as M. This symbolic representation allows us to translate the given information into mathematical expressions. We have two key pieces of information: the total number of fruits and the relationship between the number of apples and mangoes. The first piece of information tells us that the sum of apples and mangoes is 540. This can be written as an equation: A + M = 540. This equation forms the foundation of our solution, linking the two unknowns in a single, coherent statement.

The second piece of information provides a relationship between the number of apples and mangoes. It states that “Two times the number of apples is 120 more than the number of mangoes.” Translating this into an equation requires careful attention to the wording.