Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following are true if \(a \leq b ?\) (a) \(a^{2} \leq a b\) (b) \(a-3 \leq b-3\) (c) \(a^{3} \leq a^{2} b\) (d) \(-a \leq-b\)

Step-by-Step Solution

Answer

Statements (a), (b), and (c) are true.

Step 1: Analyze Inequality (b)

The statement given in part (b) is \(a - 3 \leq b - 3\). Since we know \(a \leq b\), subtracting 3 from both sides of the inequality preserves the inequality. Thus \(a - 3 \leq b - 3\) is true.

Step 2: Analyze Inequality (d)

The statement in part (d) is \(-a \leq -b\). If \(a \leq b\), then by multiplying both sides of the inequality by -1, the inequality reverses. So, \(-a \geq -b\) is true, not \(-a \leq -b\). Thus, the statement in (d) is false.

Step 3: Analyze Inequality (a)

The statement in part (a) is \(a^2 \leq ab\). We rewrite this as \(a^2 - ab \leq 0\), or \(a(a-b) \leq 0\). Since \(a \leq b\), the expression \(a-b\) is non-positive, making \(a(a-b)\) non-positive. Therefore, statement (a) is true.

Step 4: Analyze Inequality (c)

The statement in part (c) is \(a^3 \leq a^2 b\). We rewrite this as \(a^3 - a^2b \leq 0\) or \(a^2(a-b) \leq 0\). For \(a^2(a-b)\) to be true, \(a-b\) must be non-positive as before, and so it depends on \(a^2\). Since \(a^2\) is always non-negative for real \(a\), \(a^2(a-b)\) is non-positive, and hence the statement in (c) is true.

Key Concepts

Inequality Properties

Understanding inequality properties is essential when dealing with mathematical expressions. Inequalities involve statements like "less than," "greater than," or "equal to," and each of these has particular properties that help us manipulate and solve these inequalities.

• **Addition/Subtraction Property**: You can add or subtract the same number from both sides of the inequality without changing its overall sense. For instance, if you have \(a \leq b\), then \(a - 3 \leq b - 3\) remains true because you've simply subtracted 3 from both sides.
• **Multiplication/Division Property**: If you multiply or divide both sides of an inequality by a positive number, the inequality direction remains the same. However, multiplying or dividing by a negative number reverses the inequality direction. So from \(a \leq b\), when you multiply by \(-1\), you switch direction, resulting in \(-a \geq -b\).

These properties allow you to alter inequalities to simpler forms that are easier to analyze and solve.

Polynomial Inequalities

Polynomial inequalities are inequalities that contain polynomial expressions. These can sometimes take a bit more time to solve than linear inequalities due to their complexity. When approaching these inequalities, certain techniques can help simplify and understand them better.

• For example, converting inequalities into the form \(a(a-b) \leq 0\) is a critical step. It involves factoring and analyzing each part carefully.
• Consider the inequality \(a^2 \leq ab\). We can rewrite this as \(a^2 - ab \leq 0\), and factor it to become \(a(a-b) \leq 0\). Here, since \(a \leq b\) implies \(a-b\) is non-positive, the product \(a(a-b)\) is also non-positive, indicating the original inequality holds true.

By breaking down complex inequalities into simpler polynomial forms, we can logically reason through them without losing track of their original meaning.

Mathematical Reasoning

Mathematical reasoning is the skill that allows us to make sense of equations, inequalities, and other mathematical relationships. It helps convert mathematical situations into simpler, manageable forms.

• **Logical deduction**: When provided with certain conditions, like \(a \leq b\), mathematical reasoning helps apply existing mathematical properties (like inequality properties or arithmetic operations) to deduce conclusions logically. For example, knowing how multiplying by a negative impacts inequality lets us correctly interpret expressions.
• **Analysis of expressions**: To solve complex inequalities like \(a^2(a-b) \leq 0\) from the inequality \(a^3 \leq a^2b\), we use logical reasoning to factorize and examine each part of the expression. We need to ensure that our interpretation respects the mathematical context provided.

By sharpening our reasoning skills, we become more confident in navigating through mathematical challenges that can initially seem formidable.