Problem 44 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are real numbers other than 0 and \(a>b\). State whether the inequality is true or false. $$ -a<-b $$

Step-by-Step Solution

Answer

The inequality \(-a < -b\) is true, based on the given conditions \(a \neq 0\), \(b \neq 0\), and \(a > b\).

Step 1: Recall the properties of inequalities

An important property of inequalities to keep in mind is that when you multiply or divide both sides of an inequality by a negative number, the inequality symbol flips. In other words, if \(x > y\), then \((-1)x < (-1)y\), and vice versa.

Step 2: Analyze the given inequality

We are given the inequality \(a > b\), and our goal is to determine if the inequality \(-a < -b\) is true or false.

Step 3: Apply the properties of inequalities

According to the property mentioned in Step 1, when we multiply both sides of the inequality \(a > b\) by a negative number, in particular, \(-1\), the inequality symbol should flip. Thus, we have: \((-1)a < (-1)b\) This simplifies to: \(-a < -b\) This is exactly the inequality we want to prove.

Step 4: Conclusion

Based on the properties of inequalities applied to the given inequality \(a > b\), we can conclude that the inequality \(-a < -b\) is true, as long as the conditions \(a \neq 0\), \(b \neq 0\), and \(a > b\) are satisfied.

Key Concepts

Properties of Inequalities

Understanding the properties of inequalities is fundamental in solving mathematical problems that involve comparing values. One key property is the rule of sign inversion: when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol flips. So, if you start with an inequality like \(a > b\), and you multiply both sides by -1, you get \(-a < -b\). This property is also applicable to the subtraction of inequalities, as subtracting is equivalent to adding a negative number.

Another property that's sometimes overlooked yet critical in solving inequalities, is the transitive property. If \(a > b\) and \(b > c\), then it must follow that \(a > c\). This allows us to compare values indirectly, via a common third value. These principles ensure consistency within the arithmetic framework, helping students to work through complex inequalities step-by-step.

Mathematical Proof

Mathematical proof is a logical argument that verifies the truth or falsehood of a given statement. To prove the inequality \(-a < -b\) from the original exercise, we rely on deductive reasoning, starting from known truths, or axioms, and applying logical deductions. The process is a structured pathway from hypotheses—assumed starting points—to a conclusively demonstrated statement.

In our case, the hypothesis is the fact that \(a > b\). From this, we apply the property of inequalities, specifically the sign inversion rule, to reach our conclusion. The transition from hypothesis to conclusion leaves no gap or doubt, allowing the exercise to serve not only as a mathematical problem but also as an example of how mathematical proof operates in practice.

Applied Mathematics

Applied mathematics is the use of mathematical methods by different fields, such as engineering, science, business, and industry. Problems in applied mathematics often include real-world scenarios where inequalities play a significant role. Understanding and solving inequalities is crucial in forecasting, optimization, and risk assessment.

For example, consider an engineer determining load limits for materials; inequalities would be used to ensure that the design accounts for maximum and minimum tolerances. In finance, inequalities might govern investment strategies—assessing whether the potential profit from an investment outweighs the risk. The significance of inequalities in applied mathematics is not just in solving textbook exercises, but in facilitating conclusions and decisions in complex, real-life situations.