Solving Inequalities: A Step-by-Step Guide

Introduction

Hey guys! Let's dive into this interesting mathematical inequality! We have the expression (x+1)/3 + (x+4)/6 ≥ (2x^2-1)/3 - 3, and our mission, should we choose to accept it, is to find the values of x that satisfy this inequality. This involves a bit of algebraic manipulation, some simplification, and perhaps a touch of quadratic equation solving. Don’t worry, we'll break it down step by step so it’s super easy to follow. The beauty of math lies in its structured approach; by following clear steps, we can unravel even the most complex-looking problems. So, grab your pencils, maybe a cup of coffee, and let’s get started! Understanding how to solve inequalities like this is crucial not just for math class, but also for various real-world applications where constraints and limitations come into play. Whether you’re optimizing a budget, designing a structure, or even planning a road trip, the principles of inequalities can help you make informed decisions. Remember, the journey of a thousand miles begins with a single step, and the journey of solving a complex problem begins with a clear understanding of the basics. So let’s nail those basics and then watch as this inequality crumbles before our might!

Step-by-Step Solution

1. Clearing the Fractions

Okay, so the first thing we wanna do when we see fractions hanging out in an equation or inequality is to get rid of them. Fractions can make things look a bit messy, and simplifying them early on can save us a headache later. To eliminate these fractions, we need to find the least common multiple (LCM) of the denominators. In our inequality, the denominators are 3 and 6. The LCM of 3 and 6 is 6. This means we're going to multiply both sides of the inequality by 6. Remember, whatever we do to one side of an inequality, we must do to the other side to maintain the balance. It's like a mathematical see-saw; if you add weight to one side, you need to add the same weight to the other to keep it level. When multiplying by the LCM, make sure to distribute it correctly to each term. This is a common area where mistakes can happen, so let’s take it slow and be meticulous. By clearing the fractions, we transform our inequality into a more manageable form, one that's easier on the eyes and the brain. It's like decluttering your workspace before starting a big project; a clean and organized environment makes the task at hand much less daunting. So let’s multiply through by 6 and watch those fractions disappear!

Here’s how it looks:

6 * [(x+1)/3 + (x+4)/6] ≥ 6 * [(2x^2-1)/3 - 3]

This simplifies to:

2(x+1) + (x+4) ≥ 2(2x^2-1) - 18

2. Expanding and Simplifying

Alright, we've banished the fractions, which is a victory in itself! Now, let's expand and simplify the inequality. This basically means we're going to distribute any multiplication over parentheses and then combine like terms. Think of it like organizing your closet: you group similar items together to make everything neat and accessible. We'll start by distributing the constants outside the parentheses to the terms inside. This involves multiplying each term inside the parentheses by the constant factor. Remember the distributive property: a(b + c) = ab + ac. It's a fundamental tool in algebra, and we'll be using it extensively here. After distributing, we'll have a series of terms, some involving x squared, some involving x, and some constant terms. Our next step is to combine the like terms. This means adding or subtracting terms that have the same variable and exponent. For example, we can combine 2x and x because they both involve x to the power of 1. Similarly, we can combine constant terms like 2 and 4. By expanding and simplifying, we're essentially tidying up the inequality, making it easier to see the underlying structure and prepare for the next steps in solving it. It's like taking all the scattered pieces of a puzzle and sorting them into groups based on their colors and shapes; it makes the task of putting the puzzle together much more manageable.

Expanding gives us:

2x + 2 + x + 4 ≥ 4x^2 - 2 - 18

Combining like terms, we get:

3x + 6 ≥ 4x^2 - 20

3. Rearranging into a Quadratic Inequality

Now that we've simplified things, it's time to get our inequality into a standard form that we can work with more easily. Specifically, we want to rearrange it into the form ax^2 + bx + c ≤ 0 (or ≥ 0, depending on the inequality). This is the standard form for a quadratic inequality, and it allows us to use techniques like factoring or the quadratic formula to find the solutions. To achieve this, we'll move all the terms to one side of the inequality, leaving zero on the other side. This involves adding or subtracting terms from both sides, just like we did when solving regular equations. The goal is to group all the terms involving x squared, x, and constants on one side, so we can clearly see the quadratic expression. When moving terms across the inequality sign, remember to change their signs. A positive term becomes negative, and vice versa. Once we have the inequality in standard form, we'll be able to identify the coefficients a, b, and c, which are crucial for solving the quadratic inequality. Rearranging into standard form is like setting the stage for the final act; it prepares us for the climactic solution where we find the values of x that satisfy the original inequality. So, let's rearrange those terms and get ready for the grand finale!

Let's move all terms to the right side:

0 ≥ 4x^2 - 3x - 26

Or, equivalently:

4x^2 - 3x - 26 ≤ 0

4. Solving the Quadratic Inequality

Here comes the exciting part: solving the quadratic inequality! We've got our inequality in the form ax^2 + bx + c ≤ 0, and now we need to find the values of x that make this statement true. There are a couple of ways we can tackle this, and the best approach often depends on the specific quadratic expression we're dealing with. One common method is to try factoring the quadratic expression. If we can factor it into two linear expressions, we can then find the roots (or zeros) of the quadratic, which are the values of x that make the expression equal to zero. These roots are crucial because they divide the number line into intervals, and the sign of the quadratic expression will be constant within each interval. If factoring doesn't seem straightforward, we can always turn to the quadratic formula. The quadratic formula is a powerful tool that gives us the roots of any quadratic equation, regardless of whether it can be factored easily. It's a bit like having a universal key that unlocks the solutions to any quadratic puzzle. Once we have the roots, whether from factoring or the quadratic formula, we can use them to determine the intervals where the quadratic expression is less than or equal to zero (or greater than or equal to zero, depending on the inequality). This often involves creating a sign chart or testing values within each interval to see if they satisfy the inequality. Solving the quadratic inequality is like the final step in a treasure hunt; we've followed all the clues, navigated the obstacles, and now we're ready to unearth the solution that satisfies our mathematical quest.

First, let's try to factor the quadratic. We're looking for two numbers that multiply to 4 * -26 = -104 and add up to -3. After some thought, we find that 8 and -13 fit the bill. So, we can rewrite the middle term:

4x^2 - 13x + 8x - 26 ≤ 0

Now, factor by grouping:

x(4x - 13) + 2(4x - 13) ≤ 0

(x + 2)(4x - 13) ≤ 0

The roots are x = -2 and x = 13/4.

5. Determining the Solution Set

We've found the roots of the quadratic, which are like the critical points on a map. These points divide the number line into intervals, and within each interval, the sign of the quadratic expression remains consistent. Now, we need to figure out which of these intervals satisfy our inequality, which is (x + 2)(4x - 13) ≤ 0. To do this, we can use a sign chart. A sign chart is a visual tool that helps us track the sign of each factor (x + 2) and (4x - 13) within each interval. We list the roots on the number line, and then for each interval, we determine whether each factor is positive or negative. The product of the factors will then tell us the sign of the entire quadratic expression. Another way to determine the solution set is to test a value within each interval. We choose a test value that's not a root and plug it into the quadratic expression. If the result satisfies the inequality, then the entire interval is part of the solution. If the result doesn't satisfy the inequality, then the interval is not part of the solution. We repeat this process for each interval until we've identified all the intervals that satisfy the inequality. Remember, since our inequality includes “less than or equal to,” the roots themselves are also part of the solution set. Determining the solution set is like putting the final pieces of the puzzle together; we've found the critical points, and now we're connecting them to form the complete solution to our inequality.

Now we analyze the intervals:

• For x < -2, both (x + 2) and (4x - 13) are negative, so their product is positive.
• For -2 < x < 13/4, (x + 2) is positive and (4x - 13) is negative, so their product is negative.
• For x > 13/4, both (x + 2) and (4x - 13) are positive, so their product is positive.

Since we want the expression to be less than or equal to zero, the solution is the interval where the product is negative or zero. Thus, the solution is:

-2 ≤ x ≤ 13/4

Conclusion

Woohoo! We did it! We've successfully navigated the twists and turns of this inequality and arrived at our solution. By methodically clearing fractions, simplifying, rearranging into standard form, and then factoring (or using the quadratic formula), we were able to find the values of x that satisfy the inequality. Remember, solving inequalities is like solving puzzles; each step builds upon the previous one, and with a bit of patience and persistence, we can unlock the solution. The solution we found, -2 ≤ x ≤ 13/4, represents a range of values for x that make the original inequality true. This means that any number within this interval, including -2 and 13/4, will satisfy the inequality. But what does this actually mean? Inequalities are used everywhere in the real world, from figuring out the cheapest way to ship a package to designing bridges that can withstand certain loads. So, understanding how to solve them isn't just a math skill; it's a life skill! We started with a seemingly complex expression and broke it down into manageable steps, demonstrating the power of a structured approach to problem-solving. So next time you encounter an inequality, remember the steps we've covered here, and tackle it with confidence! Great job, everyone!