Problem 86 - Algebra A Combined Function

Question

If both sides of the inequality \(-3 x<-30\) are divided by \(3,\) do you reverse the direction of the inequality symbol? Why or why not?

Step-by-Step Solution

Answer

No, you do not reverse the inequality because you are dividing by a positive number.

Step 1: Addressing the Division Rule

When dividing or multiplying both sides of an inequality by a positive number, the direction of the inequality remains unchanged. Conversely, if dividing or multiplying by a negative number, the inequality sign is reversed. This step is key to solving the given problem.

Step 2: Analyze the Operation

In the inequality \(-3x < -30\), we are considering dividing both sides by \(3\). Notice that \(3\) is a positive number. According to the rule, dividing by a positive number will not change the direction of the inequality.

Step 3: Perform the Division

Divide each side of the inequality \(-3x < -30\) by \(3\). This results in \(-x < -10\). Since the division was by a positive number, the inequality direction remains the same.

Key Concepts

Inequality Rules

Inequalities are just like equations but with a twist—they involve symbols that show a relationship between two values, like less than, greater than, or not equal. Here are some important rules to remember when working with inequalities:

• **Consistency Matters:** The inequality symbol (like < or >) expresses a consistent relationship. When adding or subtracting the same number from both sides, the inequality symbol remains unchanged.
• **Multiplying/Dividing by Positives:** If you multiply or divide both sides of an inequality by a positive number, the inequality's direction does not change. For example, if you have 2 < 5 and multiply both sides by 4, you get 8 < 20, still true!
• **Handling Negatives Carefully:** In contrast, when multiplying or dividing by a negative number, the inequality symbol flips to maintain the true relationship. So, if you have -2 < 5 and multiply by -3, it changes to 6 > -15.

Understanding these rules ensures you manipulate inequalities correctly while preserving their meaning.

Positive Divisor

In the original exercise, we encounter the need to divide both sides of the inequality \(-3x < -30\) by 3. It is crucial here to note that 3 is a positive number. Let's dive into why this matters:

Dividing by a positive number keeps the order unchanged. For instance, think of 5 < 10. Dividing both sides by a positive number like 2 results in 2.5 < 5, which remains true.

• **Maintains Order:** The direction of the inequality sign stays the same because positive numbers don't flip the signs.
• **Conclusion with our Example:** After dividing \(-3x < -30\) by 3, we end up with \(-x < -10\). Since we used a positive divisor, there was no need to flip the sign—preserving the original relationship between the sides.

This reinforces the concept that operations with positive divisors are straightforward and predictable.

Negative Values

Negative values in inequalities need careful handling because they can complicate the math if not managed carefully. Let's explore why:

When negative numbers are involved, especially in multiplications or divisions, the inequality sign is reversed. This happens because negatives can disrupt the order. Imagine if you have \(-3\) and \(-9\). Generally, 3 < 9, right? But multiplying by -1 results in 3 becoming larger as \( -3 > -9 \).

• **Reversal of Relationship:** This reversal is crucial when you find negative values on both sides or are using them in operations.
• **Applying to our Problem:** While dividing \(-3x < -30\) by a positive 3 kept things straightforward, had we instead divided by \(-3\), the inequality would flip, resulting in \(x > 10\). But why? Because negative values change the direction by changing the sense of the inequality.

Understanding how negative values work within inequalities ensures you apply rules appropriately to keep equations solved accurately.