Understanding Inequalities: The Importance of Signs in Algebra

In the inequality -3x ≥ 1, what happens when you divide by a negative number?

• A. The inequality stays the same
• B. The inequality sign reverses
• C. The solution becomes negative
• D. No change occurs in the inequality

Answer: (B)

When dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is a fundamental rule in algebra that ensures the relationship remains true even after performing the operation. In the given inequality, -3x ≥ 1, if we divide both sides by -3, the sign changes from '≥' to '≤'. This results in the inequality -3x ≥ 1 becoming x ≤ -1. Thus, your selection reflects this crucial principle of inequalities, confirming that the inequality sign indeed reverses when dividing by a negative number.

Have you ever stumbled upon an inequality problem that made you scratch your head? You're not alone! Let’s break down a key concept that’s not just a math rule, but a lifesaver on tests like the FTCE General Knowledge Math. The idea of dividing by a negative number can be a little tricky, but trust me, once you get the hang of it, it can make all the difference.

So, consider the inequality -3x ≥ 1. What’s your instinct telling you here? You might think, “Dividing both sides by -3?” That’s exactly where the magic (and the challenge) happens! Here’s the kicker: when you divide or multiply both sides of an inequality by a negative number, you have to reverse the direction of the inequality sign. Sounds wild? Let me explain.

If we go ahead and divide both sides by -3, we're not just simplifying things—ah, no! We're flipping the script. The sign changes from '≥' to '≤', transforming our inequality into x ≤ -1. This is such a crucial point in algebra: the straightforward method of algebra demands we reverse the inequality to keep the equation true. Understanding this principle can really boost your confidence, especially when facing similar problems on the FTCE.

Now, why does this matter? Inequalities are everywhere—whether you're budgeting, analyzing data, or even cooking (yes, believe it or not, you might want to measure out negative amounts if you're adjusting a recipe down). The ability to manipulate them accurately is key, not just for exams but for real-life applications.

So, when you're preparing for the math section of the FTCE, take the time to practice these kinds of problems. They're not just annoying little speed bumps; they're learning tools that sharpen your logical thinking skills. And hey, every ounce of effort you put in now pays off on test day.

In essence, mastering the concept of reversing inequality signs is like fitting the final puzzle piece into a picture—it makes everything clearer and more complete. You know what? It’s moments like these that can transform your understanding of math from confusion to clarity. Just remember, every time you work with inequalities, keep that sign-reversing rule in your back pocket for success.