Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each inequality is true or false. $$-9 \geq-9$$

Step-by-Step Solution

Answer

The inequality \(-9 \geq -9\) is true.

Step 1: Understanding the inequality sign

The sign \(\geq\) signifies 'greater than or equal to'. An inequality is true whenever the expression on the left side of the inequality symbol is greater than or equal to the expression on the right side of the inequality symbol.

Step 2: Comparing values

By comparing the values on both sides of the inequality, -9 and -9, it can be seen that they are equal.

Step 3: Evaluate the inequality

Because -9 equals -9, which fits under the 'equal' part of 'greater than or equal to', the inequality \(-9 \geq -9\) is determined to be true.

Key Concepts

Inequality Symbols

In mathematics, inequality symbols are shorthand for comparing the size or order of two numbers or expressions. At the core, they tell us how one number relates to another. The most common inequality symbols include less than (<), greater than (>) , less than or equal to (≤), and greater than or equal to (≥).

For instance, the symbol ≥ is a combined symbol indicating either 'greater than' or 'equal to.' Applying this, if we see an inequality such as \( a \geq b \), it tells us that 'a' is either greater than 'b' or exactly equal to 'b.' Proper understanding of these symbols is fundamental because they are used in a wide range of mathematical areas, including algebra, calculus, and beyond. They're particularly important when solving and graphing inequalities, as they dictate the nature of the solution set.

Comparing Values in Math

Comparing values in math is a fundamental skill that involves determining the relationship between two numbers or expressions. When evaluating inequalities, such as \(-9 \geq -9\), it is crucial to compare the two values to decide if the inequality holds true.

Here are the possible outcomes when comparing values using inequality symbols:

• If we have \(a > b\), 'a' is strictly greater than 'b'.
• \(a < b\) implies that 'a' is strictly less than 'b'.
• \(a \geq b\) means 'a' is greater than or equal to 'b'.
• \(a \leq b\) indicates 'a' is less than or equal to 'b'.

Comparing numbers can involve simple direct observation, as in the given example, or more complex calculations where properties of numbers are used to deduce the relationship. Mastery of comparison is not only essential for determining the truth value of inequalities, but it also underpins much of problem-solving in algebra.

Algebraic Reasoning

Algebraic reasoning is the process of formulating, manipulating, and solving equations and inequalities based on their algebraic properties. It requires an understanding of variables, constants, expressions, and how they interact according to mathematical laws. Solving the inequality \(-9 \geq -9\) involves algebraic reasoning in recognizing that the two values are, in fact, equal, and therefore the proposition is true.

Methods of algebraic reasoning include identifying like terms, applying the distributive property, and using inverse operations to isolate variables. It also involves logically deducing outcomes, such as knowing that if 'a' equals 'b,' then any equation or inequality treating 'a' and 'b' as interchangeable holds true. Algebraic reasoning is integral to advancing in mathematics, as it forms the basis for much more complex problem-solving and provides a concrete method for testing the validity of mathematical statements.