Introduction
Hey guys! Today, we're diving deep into the world of inequalities, specifically tackling the problem −7x ≤ −3/5x − (x + 9) − 3. Inequalities might seem a bit daunting at first, but trust me, once you break them down step by step, they're totally manageable. Think of them like equations, but instead of finding a single solution, we're finding a range of solutions. This guide is designed to walk you through each stage of solving this particular inequality, explaining the logic and techniques involved. We'll cover everything from simplifying the expression to isolating the variable, and finally, expressing the solution in different forms. So, grab your pencils and notebooks, and let's get started on this mathematical journey together! Remember, the key to mastering inequalities, just like any math topic, is practice, practice, practice. The more you work through problems, the more comfortable and confident you'll become. Plus, understanding inequalities is super important for all sorts of real-world applications, from figuring out budgets to optimizing resources. So, let's jump in and unlock the secrets of solving inequalities!
Understanding Inequalities
Before we jump into solving the specific inequality at hand, it's important to understand what inequalities are and how they differ from equations. Inequalities, unlike equations, don't state that two expressions are equal. Instead, they show a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. These relationships are represented by the symbols >, <, ≥, and ≤. Think of it this way: an equation is like a perfectly balanced scale, while an inequality is like a scale that's tipped one way or the other. Now, let's talk about the properties of inequalities. These properties are the rules we use to manipulate inequalities while still maintaining their truth. One key property to remember is the multiplication/division property: when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is crucial! For example, if we have -x < 5, multiplying both sides by -1 gives us x > -5. Ignoring this flip can lead to completely wrong answers. Another important concept is the idea of a solution set. Unlike equations, which usually have one or a few specific solutions, inequalities often have a range of solutions. This range can be represented graphically on a number line, using open circles for strict inequalities (>, <) and closed circles for inclusive inequalities (≥, ≤). We can also express the solution set using interval notation, which is a concise way to show the range of values that satisfy the inequality. Understanding these basics is the foundation for tackling more complex inequalities, so make sure you're comfortable with these concepts before moving on.
Step-by-Step Solution of −7x ≤ −3/5x − (x + 9) − 3
Okay, let's get our hands dirty and solve the inequality −7x ≤ −3/5x − (x + 9) − 3 step by step. The first thing we want to do is simplify both sides of the inequality as much as possible. This makes the problem much easier to handle. On the right side, we have −3/5x − (x + 9) − 3. Let's start by distributing the negative sign inside the parentheses: −3/5x − x − 9 − 3. Next, we can combine the constant terms −9 and −3, which gives us −12. So now our inequality looks like this: −7x ≤ −3/5x − x − 12. Still looking a bit messy, right? Let's combine the x terms on the right side. We have −3/5x and −x. To combine them, we need a common denominator. We can rewrite −x as −5/5x. Adding these together, we get −3/5x − 5/5x = −8/5x. Now our inequality is much cleaner: −7x ≤ −8/5x − 12. Great! We've simplified both sides as much as we can. The next step is to isolate the x terms on one side of the inequality. To do this, let's add 8/5x to both sides. This gives us −7x + 8/5x ≤ −12. Now we need to combine the x terms on the left side. We can rewrite −7x as −35/5x. So we have −35/5x + 8/5x ≤ −12. Adding these together, we get −27/5x ≤ −12. We're almost there! The final step is to isolate x. To do this, we need to multiply both sides of the inequality by the reciprocal of −27/5, which is −5/27. But remember the golden rule of inequalities: when we multiply or divide by a negative number, we must flip the inequality sign! So, multiplying both sides by −5/27, we get x ≥ (−12) * (−5/27). Simplifying the right side, we have x ≥ 60/27. We can further simplify this fraction by dividing both the numerator and denominator by 3, which gives us x ≥ 20/9. And there you have it! Our solution is x ≥ 20/9.
Expressing the Solution
Now that we've solved the inequality and found that x ≥ 20/9, let's talk about how to express this solution in different ways. This is super important because different situations might call for different representations. The most straightforward way is the inequality notation, which we already have: x ≥ 20/9. This tells us that x can be any number that is greater than or equal to 20/9. But what does this look like on a number line? That's where the graphical representation comes in handy. To graph this solution, we draw a number line and mark the point 20/9. Since our inequality includes