Problem 35 - The Calculus with Analytic Geometry

Question

What single inequality is equivalent to the following two inequalities: \(a>b+c\) and \(a>b-c ?\)

Step-by-Step Solution

Answer

The single inequality is \(a > b + c\).

Step 1: Write Down the Given Inequalities

The problem gives two inequalities: 1. \(a > b + c\) 2. \(a > b - c\)

Step 2: Identify the Stronger Inequality

To find a single inequality that encompasses both, observe that:\(a\) must be greater than the maximum value between \(b + c\) and \(b - c\). This is the stronger condition.

Step 3: Formulate the Combined Inequality

Combining both inequalities into one, we understand that when \(a > b + c\), the condition also satisfies \(a > b - c\). Thus, the combined single inequality is \(a > b + c\).

Key Concepts

compound inequalities

Compound inequalities involve two or more inequalities that are combined into one statement with the word 'and' or 'or'. Here, the exercise deals with the concept of combining two inequalities to find a single equivalent statement. This is a common technique in algebra to simplify multiple conditions or constraints into a simpler form. For example, in the given inequalities, we have:

• \(a > b + c\)
• \(a > b - c\)

To find a single, combined inequality, we look at the strongest condition that satisfies both original inequalities. This means we need to consider the highest constraint imposed by either of the inequalities.

combining inequalities

Combining inequalities involves taking multiple inequality statements and merging them into one statement that encapsulates all the given conditions. In the exercise, we have two inequalities: \(a > b + c\) and \(a > b - c\). To merge these, we must ensure that the final inequality is stricter than both the original inequalities. We observe that if \(a\) is greater than \(b + c\), it will naturally be greater than \(b - c\) as well. This leads to the result:
\[a > b + c\]
when this is true, both original inequalities are satisfied.

algebraic expressions

Algebraic expressions involve numbers, variables, and operators. They represent mathematical relationships and can be manipulated to form equations and inequalities. In the given problem, the inequalities:

• \(a > b + c\)
• \(a > b - c\)

involve algebraic expressions with the same structure but differing signs. The goal was to combine these using algebraic manipulation to streamline the conditions into a single, stronger inequality. By understanding how to work with algebraic expressions and their properties, you can simplify and solve complex problems with ease.