Solving The Inequality: 3q - (5q + 1) > -5q - 7 - 5q

Nov 10, 2025 by Andrew McMorgan 53 views

Hey guys! Today, we're diving into a fun little math problem that involves solving an inequality. Inequalities might seem a bit intimidating at first, but trust me, they're just like equations with a twist! Instead of finding a single solution, we're looking for a range of values that make the statement true. So, let’s break down this inequality step by step and make it super easy to understand. We'll be working through the inequality $3q - (5q + 1) > -5q - 7 - 5q$ . Grab your thinking caps, and let's get started!

Understanding Inequalities

Before we jump into the nitty-gritty, let's quickly recap what inequalities are all about. Unlike equations, which use an equals sign (=), inequalities use symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Think of them as comparisons rather than strict equalities. For example, “x > 5” means that x can be any number bigger than 5, but not 5 itself. Inequalities help us define ranges or sets of possible solutions, which is super useful in many real-world scenarios. Now, let's get back to our specific problem and see how we can tackle it!

Why Inequalities Matter

Inequalities are everywhere, even if we don't always realize it. They're used in economics to model supply and demand, in computer science to optimize algorithms, and even in everyday life when we're making decisions based on constraints (like budget limits or time constraints). So, understanding inequalities isn't just about acing math tests; it's about building a valuable problem-solving skill that can be applied in countless situations. Plus, the logic and steps we use to solve inequalities are very similar to those we use for equations, so mastering inequalities can actually boost your overall math confidence! Think of it like leveling up in a video game – each concept you conquer makes you an even more formidable player.

Step 1: Simplify Both Sides of the Inequality

The first thing we want to do is simplify both sides of the inequality. This means getting rid of parentheses and combining any like terms. Remember the order of operations (PEMDAS/BODMAS)? We'll be using that here. Our inequality is: $3q - (5q + 1) > -5q - 7 - 5q$ .

Distribute the Negative Sign

First, let's deal with those parentheses. We have a negative sign in front of the (5q + 1), which means we need to distribute the negative sign to both terms inside the parentheses. This gives us:

$3q - 5q - 1 > -5q - 7 - 5q$

Notice how the + 1 inside the parentheses becomes - 1 after distributing the negative sign. This is a common mistake, so always double-check your signs! Now that we've handled the parentheses, let's move on to the next step.

Combine Like Terms

Next, we'll combine like terms on each side of the inequality. On the left side, we have 3q and -5q, which can be combined. On the right side, we have -5q and -5q, which can also be combined. Let’s do it:

Left side: $3q - 5q = -2q$

Right side: $-5q - 5q = -10q$

So, our inequality now looks like this:

$-2q - 1 > -10q - 7$

See? We've already made things much simpler! By simplifying each side, we’ve cleared the way for the next steps in solving for q. This is a crucial part of the process because it makes the inequality much easier to manipulate. Now, let's move on to isolating the variable.

Step 2: Isolate the Variable Term

Now that we've simplified both sides, our next goal is to isolate the variable term (that's the term with q in it) on one side of the inequality. To do this, we'll use inverse operations. Remember, whatever we do to one side of the inequality, we must do to the other to keep it balanced. Our simplified inequality is:

$-2q - 1 > -10q - 7$

Add 10q to Both Sides

To get all the q terms on one side, let's add 10q to both sides of the inequality. This will eliminate the -10q term on the right side:

$-2q - 1 + 10q > -10q - 7 + 10q$

Simplifying this gives us:

$8q - 1 > -7$

We're making progress! Notice how adding 10q to both sides helped us consolidate the variable terms on the left side. This is a key strategy in solving inequalities (and equations, for that matter). Now, let's get rid of that pesky constant term on the left side.

Add 1 to Both Sides

Next, we want to isolate the term with q even further, so let's get rid of the -1 on the left side. We can do this by adding 1 to both sides:

$8q - 1 + 1 > -7 + 1$

This simplifies to:

$8q > -6$

Awesome! We're getting closer and closer to our solution. We've successfully isolated the variable term on one side of the inequality. Now, the only thing left to do is to get q all by itself. Are you ready for the final step?

Step 3: Solve for the Variable

We're in the home stretch! We've simplified the inequality and isolated the variable term. Now, we just need to solve for q. Our current inequality is:

$8q > -6$

Divide Both Sides by 8

To get q by itself, we need to undo the multiplication by 8. We can do this by dividing both sides of the inequality by 8. Remember, when you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign. But since we're dividing by a positive number (8), we don't need to worry about that this time.

So, let's divide both sides by 8:

$rac{8q}{8} > rac{-6}{8}$

This simplifies to:

$q > - rac{3}{4}$

And there we have it! We've solved the inequality. The solution is q is greater than $- rac{3}{4}$ .

Final Answer and Recap

So, the solution to the inequality $3q - (5q + 1) > -5q - 7 - 5q$ is:

$q > - rac{3}{4}$

This means that any value of q that is greater than $- rac{3}{4}$ will make the original inequality true. We did it! Let’s quickly recap the steps we took to solve this inequality:

Simplify Both Sides: We distributed the negative sign and combined like terms to make the inequality easier to work with.
Isolate the Variable Term: We used inverse operations to get all the q terms on one side and the constants on the other.
Solve for the Variable: We divided both sides by the coefficient of q to find the solution.

Remember, solving inequalities is all about following these steps carefully and keeping track of your signs. With a little practice, you'll become a pro in no time! Keep up the great work, and I’ll catch you in the next math adventure!