Problem 86 - 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. $$ 5>3 $$

Step-by-Step Solution

Answer

3 < 5

Step 1: Understanding the Problem

The exercise asks us to rewrite the inequality \(5 > 3\) by reversing the inequality sign, while ensuring the meaning remains unchanged. This involves using the properties of inequalities.

Step 2: Reversing the Inequality

To reverse the inequality symbol in \(5 > 3\), swap the numbers' positions and change \(>\) to \(<\). This results in \(3 < 5\).

Step 3: Verification

Check that \(3 < 5\) conveys the same truth as \(5 > 3\). Both statements correctly express that 5 is greater than 3, just stated differently.

Key Concepts

Reversing Inequalities

Reversing an inequality means changing the direction of the inequality symbol while keeping the meaning the same. In simple terms, when you reverse an inequality, you want to make sure the statement still expresses the original relationship between the numbers involved. For example, if we start with the inequality \(5 > 3\), flipping it to \(3 < 5\) maintains the truth that 5 is greater than 3.

To reverse an inequality:

• Swap the places of the two numbers or expressions.
• Change the inequality symbol from \(>\) to \(<\) or vice versa.

Remember, simply reversing the numbers without changing the inequality symbol would alter the meaning, which we do not want. Our goal is to express the same idea in a flipped format.

Inequality Properties

When dealing with inequalities, certain properties help us manipulate and understand them without changing their meaning. Understanding these properties is key to solving and rewriting inequalities correctly. Here are some crucial properties:

• Transitive Property: If \(a > b\) and \(b > c\), then \(a > c\). This helps us chain together multiple inequalities.
• Addition/Subtraction Property: Adding or subtracting the same number on both sides of an inequality does not change the inequality, so if \(a > b\), then \(a + c > b + c\).
• Multiplication/Division Property: Multiplying or dividing both sides by a positive number keeps the sense of the inequality, e.g., if \(a > b\), changing both sides with a positive number c results in \(ac > bc\). However, be cautious when multiplying or dividing by a negative number, as it reverses the inequality sign.

It's important to use these properties correctly, as incorrect manipulations can lead to statements that no longer reflect the true relationship between quantities.

Inequality Verification

After reversing or manipulating an inequality, it's essential to verify that the new statement is logically the same as the original. Verification means checking whether the resulting statement maintains the intended comparison.

When we reversed \(5 > 3\) to \(3 < 5\), the verification process would involve ensuring both inequalities express that 5 is indeed greater than 3. Here’s how you can verify inequalities:

• Re-examine the inequality post-reversal or manipulation: see if it conveys the expected relationship.
• Test with simple calculations or substitute known numbers to check the truth of the statement.
• Recall the original context or question to ensure the reversal aligns with what was intended to be shown.

By verifying, mistakes are caught, and you gain confidence that your transformed inequality is accurate and meaningful.