Problem 24 - Algebra A Combined Function

Question

Rewrite each inequality so that the inequality symbol points in the opposite direction and the resulting statement has the same meaning as the given one. $$ 5>3 $$

Step-by-Step Solution

Answer

The inequality "5 > 3" can be rewritten as "3 < 5."

Step 1: Identify the Inequality

The inequality we have is \(5 > 3\). This states that 5 is greater than 3.

Step 2: Reverse the Inequality Symbol

The goal is to reverse the inequality symbol while retaining the same meaning. Therefore, the inequality becomes \(3 < 5\).

Step 3: Check the Results

We know \(3 < 5\) is true because 3 is indeed less than 5. This reversed version conveys the same relationship as the original statement.

Key Concepts

Reversing Inequalities

Reversing an inequality means changing the direction of the inequality symbol while keeping the statement's meaning unchanged. Consider the inequality \(5 > 3\). This tells us that 5 is greater than 3. To reverse this inequality, flip the inequality sign from '>' to '<'. Hence, the inequality will change into \(3 < 5\).

Although the sign now points to the left, the relationship between the numbers remains constant. This new inequality, \(3 < 5\), holds the same truth as the initial one because 3 is still less than 5.

It is important to understand that reversing inequalities is merely a matter of adjusting the symbol and the order of numbers; the numeric relationship itself does not change.

Inequality Symbols

Inequality symbols are vital in comparing two values. They are used to express one number being less than, greater than, less than or equal to, or greater than or equal to another number. The common symbols include:

• '>' meaning greater than
• '<' meaning less than
• '\(\geq\)' meaning greater than or equal to
• '\(\leq\)' meaning less than or equal to

Understanding these symbols is crucial as they determine the relationship between the values.

In our example of \(5 > 3\), the '>' symbol explicitly states that 5 is greater than 3. When this is reversed to \(3 < 5\), the '<' symbol conveys that 3 is less than 5, showing the same comparison from a different perspective.

Mathematical Statements

Mathematical statements, such as inequalities, are expressions that assert some fact or condition holds between numbers. These statements are foundational in math as they help explain and reason about the numeric relationships.

An inequality is a type of mathematical statement that compares two values, showing whether one is larger or smaller than the other. For instance, \(5 > 3\) is an inequality statement meaning 5 is greater than 3. By reversing the inequality to \(3 < 5\), it asserts the same condition in a different format.

Practicing with such mathematical statements enhances critical thinking and problem-solving skills since it demands understanding of number relationships and symbol meanings.