Problem 84 - Beginning and Intermediate Algebra

Question

Rewrite the following inequalities so that the inequality symbol points in the opposite direction and the resulting statement has the same meaning as the given one. $$ -13 \leq 13 $$

Step-by-Step Solution

Answer

Rewrite as \(13 \geq -13\).

Step 1: Identify the Original Inequality

The original inequality given is \(-13 \leq 13\). This means \(-13\) is less than or equal to \(13\).

Step 2: Determine the Reversed Inequality

To reverse the inequality sign in \(-13 \leq 13\), we will change the \(\leq\) to \(\geq\). This changes the direction of the inequality.

Step 3: Rewrite the Inequality

Switching the sides, we rewrite the reversed inequality as \(13 \geq -13\). This inequality has the same meaning, indicating that \(13\) is greater than or equal to \(-13\).

Key Concepts

Understanding Inequality Reversal

Inequality reversal is a key concept in algebra that allows us to rephrase an inequality while maintaining its validity. When we reverse an inequality, we switch the inequality symbol used in the expression:

• For "less than or equal to" (\( \leq \)), we use "greater than or equal to" (\( \geq \)).
• For "greater than or equal to" (\( \geq \)), we use "less than or equal to" (\( \leq \)).
• For "less than" (\( < \)), it becomes "greater than" (\( > \)).
• For "greater than" (\( > \)), it becomes "less than" (\( < \)).

By reversing the inequality, we simply ensure the relationship between the numbers or expressions still holds true. For instance, the original inequality \(-13 \leq 13\) can also be expressed as \(13 \geq -13\), preserving the same relationship but displaying it from a different perspective.

Decoding Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators that represent a certain value or relationship. In the case of inequalities, expressions involve inequality symbols (\( \leq, \geq, <, > \)) that denote relationships between numbers.

Consider the expression \(-13 \leq 13\):

• The number \(-13\), on the left, is compared to \(13\) on the right.
• The inequality operator \( \leq \) indicates that \(-13\) is less than or equal to \(13\).

When working with mathematical expressions, especially inequalities, it’s crucial to evaluate the numbers and symbols accurately to understand the true relationship they represent. This attention to detail ensures that students can confidently manipulate and solve equations or inequalities in their mathematical journey.

Exploring Inequality Symbols

Inequality symbols are the backbone of expressing hierarchical relationships in mathematics. These symbols visually demonstrate how one number or expression compares to another.

• The symbol \( \leq \), or "less than or equal to," means the value on the left is smaller than or equal to the one on the right.
• The symbol \( \geq \), or "greater than or equal to," means the value on the left is greater than or equal to the one on the right.
• "Less than" \( < \) and "greater than" \( > \) symbols only describe strict inequalities, indicating no equality is possible.

These symbols allow us to represent a range of scenarios, from simple number comparisons to complex variables within equations. Understanding and applying inequality symbols accurately enables students to solve real-world problems and interpret data relationships effectively.