X-7 X-6 Is Less Than Or Equal To Inequality, Explained In Simple Terms You Can Actually Understand

Prof. Dora Volkman Jr. 21 May 2025

Alright folks, let’s dive into something that might sound a little intimidating at first: solving the inequality x-7 x-6 ≤ 0. Don’t worry, I’m here to break it down step by step so even if math isn’t your strong suit, you’ll walk away feeling confident. This isn’t just about numbers; it’s about understanding how inequalities work and why they matter in everyday life. So buckle up, because we’re about to turn this math problem into something relatable and easy to grasp.

Now, if you’re reading this, chances are you’ve encountered this inequality somewhere—whether it’s for school, work, or just personal curiosity. The good news? It’s simpler than it looks. We’re going to tackle it head-on, breaking it into bite-sized pieces that anyone can follow. Stick with me, and by the end of this article, you’ll not only know how to solve it but also understand why it’s useful.

Before we get too far, let’s talk about why inequalities matter. They’re not just abstract concepts locked away in math textbooks. Inequalities show up everywhere—in budgeting, planning, engineering, and even figuring out how much pizza you need for a party. Understanding them means empowering yourself to make smarter decisions. So, let’s jump in and make sense of x-7 x-6 ≤ 0 together!

What the Heck Is an Inequality Anyway?

First things first, let’s clear up what an inequality actually is. Think of it as a cousin to the good old equation, but instead of saying “this equals that,” inequalities say “this is less than, greater than, less than or equal to, or greater than or equal to that.” It’s like when your mom says, “You can have fewer than three cookies,” or when your boss says, “This project needs to be done in no more than two weeks.”

In math terms, inequalities use symbols like , ≤, and ≥ to compare two expressions. And that’s exactly what we’re dealing with here: x-7 x-6 ≤ 0. It’s asking us to find all the values of x that make this statement true. But don’t panic—we’ll get to solving it soon enough.

Breaking Down the Problem: x-7 x-6 ≤ 0

Alright, let’s dissect this inequality piece by piece. We’ve got two parts here: x-7 and x-6. These are what we call linear expressions, and when multiplied together, they create what’s known as a quadratic inequality. Quadratic? Yeah, it sounds fancy, but it just means we’re dealing with an equation where the highest power of x is 2.

So, our goal is to figure out when the product of (x-7) and (x-6) is less than or equal to zero. To do that, we’ll use a technique called the sign chart, which helps us visualize where the expression changes from positive to negative. Stick with me—it’s not as complicated as it sounds!

Step 1: Finding the Critical Points

The first step in solving any inequality is finding the critical points. These are the values of x that make either part of the inequality equal to zero. In our case, we set x-7 = 0 and x-6 = 0, which gives us two solutions: x = 7 and x = 6. These points are important because they divide the number line into sections where the sign of the expression might change.

Think of it like this: if you’re driving down a road and you hit a stop sign, you pause before continuing. Similarly, these critical points are like stop signs for our inequality—they’re the places where the expression flips from positive to negative or vice versa.

Why Critical Points Matter

These points are crucial because they help us determine where the inequality holds true. Without them, we’d just be guessing blindly. By finding x = 6 and x = 7, we’ve narrowed down the possible solutions to specific ranges on the number line. It’s like setting up boundaries for our problem.

Step 2: Building the Sign Chart

Now that we’ve got our critical points, it’s time to build the sign chart. This is where things start to come together. We’ll divide the number line into three intervals: one before 6, one between 6 and 7, and one after 7. Then, we’ll test a sample value from each interval to see whether the expression is positive, negative, or zero.

Here’s how it works:

For x
For 6
For x > 7, pick a number like x = 8. Plug it into the expression: (8-7)(8-6) = (1)(2) = 2. Positive again!

See how the sign changes depending on which interval you’re in? That’s the magic of the sign chart—it helps us visualize the behavior of the inequality.

What Does the Sign Chart Tell Us?

The sign chart shows us that the expression is negative only when 6

Solving the Inequality

Putting it all together, the solution to x-7 x-6 ≤ 0 is the interval [6, 7]. This means that any value of x between 6 and 7, including 6 and 7 themselves, satisfies the inequality. It’s like saying, “If you’re driving between mile markers 6 and 7, you’re good to go.”

But what does this mean in practical terms? Well, inequalities like this often pop up in real-world scenarios. For example, imagine you’re trying to figure out how much money you can spend on groceries without going over budget. Or maybe you’re planning a road trip and need to know how far you can drive before running out of gas. Inequalities help you set boundaries and make informed decisions.

Why Understanding Inequalities Is Important

Now that we’ve solved the inequality, let’s talk about why this stuff matters. Inequalities aren’t just for math class—they’re tools for solving real-life problems. Whether you’re balancing your checkbook, optimizing a business plan, or designing a roller coaster, inequalities help you stay within limits and avoid disaster.

Plus, understanding inequalities boosts your critical thinking skills. It teaches you to analyze problems systematically, break them into manageable parts, and arrive at logical conclusions. And let’s be honest—who doesn’t love feeling smart and capable?

Real-World Applications of Inequalities

Here are a few examples of how inequalities show up in everyday life:

Finance: Inequalities help you determine how much you can afford to spend without overspending.
Engineering: Engineers use inequalities to ensure structures are safe and stable under varying conditions.
Healthcare: Doctors use inequalities to calculate dosages and ensure treatments are effective without being harmful.
Education: Teachers use inequalities to set grading scales and ensure fairness in assessments.

See? Inequalities are everywhere, and mastering them gives you a powerful tool for navigating the world.

Common Mistakes to Avoid

Before we wrap up, let’s talk about some common pitfalls people fall into when solving inequalities. Avoiding these mistakes will save you time and frustration:

Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
Ignoring the critical points and skipping the sign chart step.
Not including the endpoints when the inequality says “less than or equal to” or “greater than or equal to.”
Overcomplicating the problem by trying to solve it all at once instead of breaking it into steps.

Remember, math is all about patience and precision. Take your time, follow the steps, and double-check your work. You’ll be amazed at how much easier it becomes.

How to Check Your Work

Once you’ve solved an inequality, it’s always a good idea to check your solution. Plug a few values from the solution set back into the original inequality to make sure they work. For example, try x = 6, x = 6.5, and x = 7 in x-7 x-6 ≤ 0. If they all satisfy the inequality, you’re good to go!

Final Thoughts and Takeaways

And there you have it—a complete breakdown of how to solve the inequality x-7 x-6 ≤ 0. By finding the critical points, building a sign chart, and analyzing the intervals, we determined that the solution is [6, 7]. But more importantly, we explored why inequalities matter and how they apply to real life.

So, here’s what you need to remember:

Inequalities compare expressions using , ≤, or ≥.
Critical points help divide the number line into intervals.
The sign chart shows where the expression is positive, negative, or zero.
Inequalities are useful tools for setting boundaries and solving real-world problems.

Now it’s your turn! Try solving a few more inequalities on your own, and don’t hesitate to reach out if you have questions. Math is all about practice, and the more you do, the better you’ll get. And who knows? You might even start enjoying it!

Call to Action

Did you find this article helpful? Let me know in the comments below! Share it with your friends, and check out some of our other math-related content while you’re here. Together, we’ll make math less scary and more approachable—one inequality at a time. Cheers!

What the Heck Is an Inequality Anyway?
Breaking Down the Problem: x-7 x-6 ≤ 0
Step 1: Finding the Critical Points
Step 2: Building the Sign Chart
Solving the Inequality
Why Understanding Inequalities Is Important
Common Mistakes to Avoid
Final Thoughts and Takeaways
Call to Action

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