Problem 83 - 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. $$ 25 \geq 20 $$

Step-by-Step Solution

Answer

20 \leq 25

Step 1: Understanding the Problem

The inequality given is \( 25 \geq 20 \). This means 25 is greater than or equal to 20. Our task is to rewrite this inequality such that the symbol points in the opposite direction while maintaining the original meaning.

Step 2: Rewriting the Inequality Symbol

The original inequality symbol is \( \geq \), which points towards 20. To change the direction, we switch it to \( \leq \), so it will point towards 25 in the rewritten inequality.

Step 3: Switching the Sides

When we switch the sides of 25 and 20, the inequality becomes \( 20 \leq 25 \), maintaining the same relationship meaning. The statement now reads as 20 is less than or equal to 25.

Key Concepts

Inequality Symbols

Inequality symbols are crucial for comparing two numbers, indicating how one value relates to another. The four most common inequality symbols are:

• \( > \) means "greater than"
• \( < \) means "less than"
• \( \geq \) means "greater than or equal to"
• \( \leq \) means "less than or equal to"

Each symbol helps define the relationship between two values. For example, \( 25 \geq 20 \) tells us that 25 is greater than or equal to 20.
This is useful in various scenarios, such as determining allowed limits and ranges in real-life problems. Understanding these symbols forms the basis for writing and solving inequalities accurately.

Rewriting Inequalities

Rewriting inequalities is a skill needed to manipulate these mathematical statements while keeping their meaning intact. When we rewrite an inequality, we often aim to either simplify it or to change its structure for better understanding.
To rewrite an inequality, you can use basic mathematical principles. If the original inequality is \( 25 \geq 20 \), we retain the inherent relationship but can present it differently.
For example, if we need the inequality symbol to point the other way, we need to ensure the relationship remains valid. This might mean switching the symbol to \( \leq \), as in our exercise, because while reversing the inequality, the math holds: \( 20 \leq 25 \).
Mastering this concept allows students to flexibly approach problems, often leading to simpler or more comprehensible forms of the same mathematical reality.

Switching Sides in Inequalities

Switching sides in inequalities involves reversing the position of numbers or expressions around the inequality symbol. This can be both a helpful visual aid and a useful technique for solving or analyzing inequalities.By switching the sides, as in turning \( 25 \geq 20 \) into \( 20 \leq 25 \), the inequality still holds true. Here's how it works:

• Identify the original relationship, for example, 25 is more than 20, represented by \( \geq \)
• Swap the numbers or terms on either side of the inequality
• Change the inequality symbol accordingly. If switching sides, \( \geq \) becomes \( \leq \)

This operation maintains the logical statement's authenticity.
It's important to practice switching sides to become comfortable with analyzing inequalities flexibly.
This technique is especially useful in algebra when solving for unknowns within inequalities.