Solving Inequalities $2(x-3) \geq -3(-3+x)$ Step-by-Step Guide

August 6, 2025 by Scholario Team 63 views

$Solving the Inequality: $2(x-3) ${\geq}$ -3(-3+x)$$

Hey guys! Today, let's dive into solving a linear inequality. It's like solving an equation, but with a little twist because we're dealing with greater than or equal to ( $\geq$ ) signs instead of just equals (=). Don't worry, it's totally manageable. We'll break it down step-by-step so you can ace these types of problems. Our specific problem is $2(x-3) \geq -3(-3+x)$ , and we need to figure out which of the given options is the correct solution. The options are:

A. $x \geq 3$ B. $x \leq 3$ C. $x \leq 2$ D. $x \geq 2$

So, let's get started and find the solution together!

Step-by-Step Solution

The key to solving inequalities is to isolate the variable, just like when you're solving equations. We'll use algebraic manipulations, keeping in mind that multiplying or dividing by a negative number flips the inequality sign. Let's get started, guys!

1. Distribute on Both Sides

The first thing we need to do is get rid of those parentheses. We'll distribute the numbers outside the parentheses to the terms inside. Remember the distributive property? It's where you multiply the term outside the parenthesis by each term inside the parenthesis. For our inequality, $2(x-3) \geq -3(-3+x)$ , we have:

On the left side: $2 * x = 2x$ and $2 * -3 = -6$ . So, $2(x-3)$ becomes $2x - 6$ .
On the right side: $-3 * -3 = 9$ and $-3 * x = -3x$ . So, $-3(-3+x)$ becomes $9 - 3x$ .

Now our inequality looks like this: $2x - 6 \geq 9 - 3x$ . Awesome! We've gotten rid of the parentheses and can move on to the next step.

2. Combine Like Terms

Next, we want to get all the terms with x on one side of the inequality and the constant terms on the other side. This is just like solving a regular equation. Let's move the $-3x$ term from the right side to the left side. To do this, we'll add $3x$ to both sides of the inequality. This keeps the inequality balanced. So we have:

$2x - 6 + 3x \geq 9 - 3x + 3x$

Simplifying this, we get:

$5x - 6 \geq 9$

Great! Now we have all the x terms on the left side. Let's move the constant term, $-6$ , to the right side. To do this, we'll add 6 to both sides:

$5x - 6 + 6 \geq 9 + 6$

This simplifies to:

$5x \geq 15$

We're almost there, guys! We've combined the like terms and now we just need to isolate x.

3. Isolate the Variable

To isolate x, we need to get rid of the 5 that's multiplying it. We'll do this by dividing both sides of the inequality by 5. Since 5 is a positive number, we don't need to flip the inequality sign. Here we go:

$\frac{5x}{5} \geq \frac{15}{5}$

This simplifies to:

$x \geq 3$

Boom! We've done it! We've solved the inequality and found that $x$ is greater than or equal to 3.

Identifying the Correct Option

Now that we've solved the inequality, let's look back at the options and see which one matches our solution:

A. $x \geq 3$ This is our solution! B. $x \leq 3$ C. $x \leq 2$ D. $x \geq 2$

So, the correct answer is A. $x \geq 3$ . Awesome job, guys! We solved it together.

Key Concepts Revisited

Let's quickly recap the key concepts we used to solve this inequality. Understanding these concepts will help you tackle similar problems with confidence. Inequalities, at their core, are about relationships that aren't strictly equal. They express a range of possible values rather than a single one. We use symbols like < (less than), > (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to) to represent these relationships. The beauty of inequalities is that they model real-world situations where things aren't always exact, such as budgets, temperature ranges, or speed limits.

Distribution

The distributive property is super important when dealing with expressions inside parentheses. It allows us to multiply a term outside the parentheses by each term inside, effectively expanding the expression. Remember, it's like sharing the love (or the multiplication) with everyone inside the group (parentheses).

Combining Like Terms

Combining like terms is all about simplifying expressions. We group together terms that have the same variable and exponent, and we combine the constant terms as well. This makes the inequality easier to work with and helps us isolate the variable we're trying to solve for. It's like organizing your toolbox—putting the wrenches with the wrenches and the screwdrivers with the screwdrivers.

Isolating the Variable

The ultimate goal in solving any inequality (or equation) is to isolate the variable. This means getting the variable by itself on one side of the inequality. We do this by performing inverse operations (addition/subtraction, multiplication/division) on both sides, always keeping the inequality balanced. Imagine it like peeling an onion—you remove the layers one by one until you get to the core (the variable).

Flipping the Inequality Sign

This is a crucial rule to remember: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers on the number line. Think of it like looking in a mirror—everything is reversed. For example, if 2 < 4, then multiplying both sides by -1 gives -2 > -4.

Common Mistakes to Avoid

Solving inequalities can be tricky, and it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for, guys!

Forgetting to Distribute Properly

When you have a number multiplied by an expression in parentheses, make sure you distribute it to every term inside the parentheses. It's easy to forget one, especially if there are multiple terms. Double-check your work to make sure you've distributed correctly.

Not Flipping the Sign

The most common mistake is forgetting to flip the inequality sign when multiplying or dividing by a negative number. Always remember this rule, and make it a habit to double-check your steps whenever you multiply or divide by a negative. Maybe even write yourself a little reminder on your paper!

Incorrectly Combining Like Terms

Make sure you're only combining terms that are actually "like" – that is, they have the same variable and exponent. Don't try to combine an x term with a constant term, for example. Think of it like adding apples and oranges—they're both fruit, but you can't combine them into a single category.

Making Arithmetic Errors

Simple arithmetic errors can derail your entire solution. Take your time, double-check your calculations, and don't rush through the steps. It's better to be slow and accurate than fast and wrong. We all make mistakes, but careful checking can catch them before they cause problems.

Misinterpreting the Inequality Sign

Be sure you understand what the inequality signs mean. $x > 3$ means x is greater than 3, but it doesn't include 3. $x \geq 3$ means x is greater than or equal to 3, so it does include 3. Pay attention to the details!

Practice Problems

Okay, guys, now it's your turn to put what you've learned into practice! Solving inequalities becomes much easier with practice. Here are a few problems you can try on your own. Remember to follow the steps we discussed, and don't be afraid to make mistakes – that's how we learn! Grab a pencil and paper, and let's get started.

Solve the inequality: $3(x + 2) < 5x - 4$
Solve the inequality: $-2(x - 1) \geq 4x + 6$
Solve the inequality: $\frac{1}{2}x + 3 \leq 2x - 1$

Hints and Tips

For problem 1, remember to distribute the 3 on the left side and then combine like terms.
For problem 2, pay close attention to the negative sign in front of the 2. Remember to distribute it correctly, and don't forget to flip the inequality sign if necessary!
For problem 3, you might want to get rid of the fraction first by multiplying both sides of the inequality by 2. This will make the problem easier to work with.

Take your time, work through each step carefully, and check your answers. You can even graph your solutions on a number line to visualize the range of values that satisfy the inequality. Inequalities are a fundamental concept in algebra, so mastering them now will set you up for success in more advanced math courses.

Real-World Applications

You might be wondering, guys, "When will I ever use this stuff in the real world?" Well, inequalities are actually used all the time in various fields. Let's look at a few examples:

Budgeting

When you're creating a budget, you often have constraints. For example, you might want to spend less than or equal to a certain amount on groceries each week. This can be represented with an inequality. Let's say you want to spend no more than $100 on groceries. If x represents the amount you spend, the inequality would be $x \leq 100$ .

Speed Limits

Speed limits are a classic example of inequalities. The speed you drive must be less than or equal to the posted speed limit. If the speed limit is 65 mph, and s represents your speed, the inequality would be $s \leq 65$ .

Temperature Ranges

Temperature ranges are often expressed using inequalities. For example, you might want to know the days when the temperature will be between 70 and 80 degrees Fahrenheit. If t represents the temperature, the inequality would be $70 < t < 80$ .

Manufacturing

In manufacturing, inequalities are used to ensure quality control. For example, a machine might need to produce parts that are within a certain tolerance range (e.g., the diameter of a bolt must be between 1.95 cm and 2.05 cm). These tolerances can be expressed as inequalities, ensuring that the manufactured products meet the required specifications.

Dosage Calculations

In medicine, inequalities are crucial for calculating dosages. Doctors need to ensure that the amount of medication given is within a safe and effective range. The dosage must be greater than or equal to a minimum effective dose and less than or equal to a maximum safe dose.

Resource Allocation

In business and economics, inequalities are used to model resource allocation problems. For example, a company might have a limited budget and need to decide how to allocate it among different projects. Inequalities can help determine the optimal allocation strategy, ensuring that the available resources are used efficiently.

Computer Science

In computer science, inequalities are used in algorithms and data structures. For example, search algorithms often use inequalities to narrow down the search space and find the desired element. Inequalities are also used in optimization problems, where the goal is to find the best solution that satisfies certain constraints.

Conclusion

So, there you have it, guys! We've tackled the inequality $2(x-3) \geq -3(-3+x)$ , step by step. We distributed, combined like terms, isolated the variable, and found the solution: $x \geq 3$ . Remember the key concepts: the distributive property, combining like terms, isolating the variable, and flipping the inequality sign when multiplying or dividing by a negative number. Avoid the common mistakes, practice regularly, and you'll become a pro at solving inequalities! Inequalities might seem abstract, but they have tons of real-world applications, from budgeting to speed limits to manufacturing. Keep practicing, and you'll be amazed at how useful these skills can be. Keep up the awesome work, guys!