Solving Inequalities: A Step-by-Step Guide

Let's dive into the world of inequalities! In this article, we'll break down how to solve the inequality $2(4x - 3) \geq -3(3x) + 5x$. Inequalities might seem daunting at first, but with a systematic approach, they become quite manageable. We will walk through each step, ensuring you understand the logic and mechanics involved. By the end, you'll not only have the solution but also a solid understanding of how to tackle similar problems. Understanding inequalities is crucial in various fields, from mathematics and physics to economics and computer science. This guide aims to make the process clear and accessible, so you can confidently solve these types of problems.

Understanding Inequalities

Before we jump into the solution, let's establish a solid foundation. Inequalities, unlike equations, don't deal with strict equality. Instead, they express relationships where one value is greater than, less than, greater than or equal to, or less than or equal to another value. The symbols we use to represent these relationships are:

• $>$ (greater than)
• $<$ (less than)
• $\geq$ (greater than or equal to)
• $\leq$ (less than or equal to)

Think of it like a balancing scale, but instead of perfectly balanced weights, one side is heavier or lighter than the other. When we solve an inequality, we're essentially trying to isolate the variable on one side to determine the range of values that satisfy the relationship. This is a fundamental concept in algebra and is used extensively in various real-world applications. For instance, inequalities can help determine the maximum load a bridge can handle, the minimum profit a business needs to make, or the range of acceptable temperatures for a chemical reaction. The key takeaway here is that understanding inequalities is about grasping the concept of a range of possible solutions rather than a single, fixed value.

Basic Rules for Solving Inequalities

Just like solving equations, inequalities follow specific rules. Most algebraic manipulations apply similarly, but there's one crucial difference: when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. Let’s recap some key principles:

1. Addition and Subtraction: You can add or subtract the same number from both sides of the inequality without changing the direction of the inequality sign.
2. Multiplication and Division by a Positive Number: You can multiply or divide both sides by the same positive number without changing the inequality sign.
3. Multiplication and Division by a Negative Number: When you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. This is the most critical rule to remember!
4. Simplification: Combine like terms on each side of the inequality to simplify the expression.

These rules are the building blocks for solving any inequality. For example, if we have the inequality $x + 3 < 7$, we can subtract 3 from both sides to get $x < 4$. This means any value of $x$ less than 4 will satisfy the inequality. Similarly, if we have $-2x > 6$, dividing both sides by -2 (and flipping the sign!) gives us $x < -3$. These basic rules, when applied correctly, will lead you to the solution. Keep these rules in mind as we proceed to solve our main problem. Remember, consistent practice with these rules is key to mastering the art of solving inequalities.

Solving the Inequality Step-by-Step

Now, let's tackle the given inequality: $2(4x - 3) \geq -3(3x) + 5x$. We will proceed step-by-step, ensuring clarity and accuracy.

Step 1: Distribute

First, we need to simplify both sides of the inequality by distributing the numbers outside the parentheses. This means multiplying the 2 by both terms inside the first set of parentheses and the -3 by the term inside the second set.

• $2 * (4x) = 8x$
• $2 * (-3) = -6$
• $-3 * (3x) = -9x$

So, after distributing, our inequality looks like this: $8x - 6 \geq -9x + 5x$

Distribution is a fundamental step in simplifying expressions, whether they are part of equations or inequalities. It allows us to remove parentheses and combine like terms, making the problem easier to manage. Imagine distribution as opening up a package; you're revealing the individual components inside. In this case, we're revealing the individual terms that make up each side of the inequality. This process is crucial for solving not just linear inequalities, but also more complex algebraic expressions. Mastering distribution is a cornerstone of algebraic manipulation, and it's a skill that will serve you well in many mathematical contexts.

Step 2: Combine Like Terms

Next, we need to combine like terms on each side of the inequality. On the right side, we have $-9x$ and $+5x$, which are both terms with the variable $x$. Combining them gives us:

• $-9x + 5x = -4x$

Now our inequality is: $8x - 6 \geq -4x$

Combining like terms is another crucial simplification step. It's like sorting through a collection of objects and grouping the similar ones together. In algebra, this means bringing together terms that have the same variable and exponent. This step helps to streamline the equation or inequality, making it easier to see the relationships between the different parts. In our case, combining $-9x$ and $5x$ into $-4x$ simplifies the right side of the inequality, bringing us closer to isolating the variable. This process isn't just useful for solving inequalities; it's a fundamental skill in algebra that applies to a wide range of problems, from simplifying polynomials to solving systems of equations.

Step 3: Move Variables to One Side

To isolate the variable $x$, we need to move all the terms containing $x$ to one side of the inequality. A common approach is to add $4x$ to both sides. This eliminates the $-4x$ term on the right side:

• $8x - 6 + 4x \geq -4x + 4x$
• $12x - 6 \geq 0$

Moving variables to one side is a key strategy in solving any equation or inequality. The goal is to group all the variable terms together so that we can eventually isolate the variable itself. It's like gathering all the puzzle pieces of the same type before you start assembling them. By adding $4x$ to both sides, we've effectively shifted the $x$ term from the right side to the left, making the inequality cleaner and more manageable. This technique is not only applicable to linear inequalities but also to more complex equations and systems of equations. Learning to manipulate equations and inequalities by moving terms around is a core algebraic skill.

Step 4: Isolate the Variable Term

Now, we want to isolate the term with $x$. To do this, we add 6 to both sides of the inequality:

• $12x - 6 + 6 \geq 0 + 6$
• $12x \geq 6$

Isolating the variable term is like peeling away the layers to get to the core. We're progressively simplifying the inequality until only the term with the variable remains on one side. Adding 6 to both sides removes the constant term from the left side, bringing us one step closer to our goal. This step is consistent with the basic rules of algebra, which allow us to perform the same operation on both sides of an equation or inequality without changing its validity. Mastering the process of isolating the variable term is essential for solving not just simple inequalities, but also more complex algebraic problems.

Step 5: Solve for x

Finally, to solve for $x$, we divide both sides of the inequality by 12:

• $12x / 12 \geq 6 / 12$
• $x \geq 0.5$

Dividing both sides by the coefficient of $x$ is the final step in isolating the variable. It's like cutting a cake into equal slices; we're dividing both sides by the same amount to maintain the balance. In this case, dividing by 12 gives us the solution for $x$. This step is a direct application of the rules of algebra, and it's a fundamental technique in solving equations and inequalities. The result, $x \geq 0.5$, tells us that any value of $x$ greater than or equal to 0.5 will satisfy the original inequality. This is the culmination of all the previous steps, and it demonstrates the power of algebraic manipulation in finding solutions to mathematical problems.

The Solution

So, the solution to the inequality $2(4x - 3) \geq -3(3x) + 5x$ is $x \geq 0.5$. This corresponds to option A. This means that any value of $x$ that is greater than or equal to 0.5 will make the inequality true.

Understanding the solution is as important as finding it. In this case, $x \geq 0.5$ represents a range of values, not just a single number. It includes 0.5 itself and every number greater than 0.5. This is a key difference between solving equations and inequalities; equations typically have a single solution (or a few discrete solutions), while inequalities have a range of solutions. Visualizing the solution on a number line can be helpful; in this case, it would be a line extending from 0.5 to the right, including 0.5 itself. Understanding this concept is crucial for applying inequalities to real-world problems, where solutions often exist within a range rather than as a single value.

Conclusion

In this article, we've walked through the step-by-step process of solving the inequality $2(4x - 3) \geq -3(3x) + 5x$. We covered distributing, combining like terms, moving variables, isolating the variable term, and finally, solving for $x$. The solution, $x \geq 0.5$, represents all values of $x$ that satisfy the original inequality. Inequalities are a fundamental concept in mathematics, with applications spanning various fields. Mastering the techniques to solve them empowers you to tackle a wide range of problems. Remember to practice regularly and apply these steps to different types of inequalities to solidify your understanding.

For further exploration and practice on inequalities, you might find resources like Khan Academy's algebra section incredibly helpful. Check out their materials on Solving Inequalities for more examples and explanations.