Proving Inequalities: A Step-by-Step Guide

Hey guys! Let's dive into the world of inequalities and figure out how to prove them. We'll break down two specific examples, making sure everything is crystal clear. So, grab your pens and let's get started! Understanding inequalities is super important in algebra, as they help us compare values and solve a variety of problems. This guide will walk you through each step, ensuring you not only understand the proofs but also become more comfortable with algebraic manipulations in general. We will explore two types of inequalities, one involving a straightforward comparison and the other dealing with quadratic expressions. By working through these examples, you'll gain a strong foundation in the principles of inequality proving, which can then be applied to more complex problems. Remember, practice makes perfect, so don't hesitate to work through these examples multiple times until you are confident with the process. We will start by addressing the first inequality and then proceed to the second, providing detailed explanations along the way.

Proving (a-6)(a+4) < (a+2)(a-4)

Alright, let's tackle our first inequality: (a-6)(a+4) < (a+2)(a-4). Our goal is to show that this inequality holds true for any value of the variable 'a'. To do this, we'll simplify both sides of the inequality separately and then compare them. This process involves expanding the products and simplifying the resulting expressions. Remember, the key here is to isolate the variable and demonstrate a clear relationship between the two sides of the inequality. This demonstration requires careful attention to the order of operations and proper handling of algebraic terms. We'll walk through each step to ensure we arrive at the correct conclusion. Always double-check your calculations to minimize errors and ensure the proof's validity. This structured approach will help us confirm the inequality, no matter the value of 'a'. By the end of this process, you'll not only understand how to solve this inequality but also gain a better grasp of algebraic techniques that apply to various math problems.

Let's start by expanding the left side of the inequality, (a-6)(a+4). When we multiply these two binomials, we get:

a² + 4a - 6a - 24

Which simplifies to:

a² - 2a - 24

Now, let's expand the right side of the inequality, (a+2)(a-4):

a² - 4a + 2a - 8

Which simplifies to:

a² - 2a - 8

So, our inequality now looks like this:

a² - 2a - 24 < a² - 2a - 8

Next, we can subtract 'a²' and '-2a' from both sides of the inequality. This gives us:

-24 < -8

As you can see, this statement is true. Since we arrived at a true statement, regardless of the value of 'a', the original inequality holds true for any value of the variable. We have successfully proven the inequality (a-6)(a+4) < (a+2)(a-4).

Remember, the critical steps here involve expanding the products, simplifying the expressions, and ensuring that the resulting statement is consistent. The consistency of the final statement confirms the initial inequality's validity for all values of 'a'.

Proving a² - 10a + 26 > 0

Now, let's prove the inequality a² - 10a + 26 > 0. This inequality involves a quadratic expression, which requires a slightly different approach. Our aim is to show that this expression is always greater than zero, no matter what value 'a' takes. This type of proof often involves completing the square, which allows us to rewrite the quadratic expression into a form that reveals its minimum value. Completing the square is a powerful technique that transforms a quadratic expression into a perfect square plus a constant. This method allows us to analyze the expression's behavior and determine its minimum or maximum value. The ultimate goal is to demonstrate that the quadratic expression is always positive, irrespective of the value of 'a'.

To do this, we'll complete the square. Here's how it goes:

First, focus on the terms with 'a': a² - 10a. We need to create a perfect square trinomial. To do this, take half of the coefficient of 'a' (-10), square it, and add it to the expression. Half of -10 is -5, and (-5)² = 25.

So, we'll rewrite our inequality, adding and subtracting 25 to keep the expression balanced:

a² - 10a + 25 + 26 - 25 > 0

Now, group the first three terms, which form a perfect square trinomial:

(a² - 10a + 25) + 1 > 0

The perfect square trinomial simplifies to (a - 5)², so we have:

(a - 5)² + 1 > 0

Now we have a perfect square plus 1. We know that any real number squared is always greater than or equal to zero. Therefore, (a - 5)² is always ≥ 0. Because (a - 5)² is at least 0, adding 1 to it will always result in a value greater than 0. So, (a - 5)² + 1 will always be greater than 0. This confirms that a² - 10a + 26 > 0 for all values of 'a'.

In this case, the quadratic expression has a minimum value that is greater than zero. This confirms our proof. The process of completing the square enables us to transform the expression and clearly demonstrate that it always remains positive. The steps of completing the square can feel tricky initially, but with practice, you'll become comfortable with the method, and you will be able to successfully prove more complex inequalities.

Conclusion

Great job, guys! We've successfully proven both inequalities. You've seen how to handle straightforward comparisons and how to work with quadratic expressions. Remember, practice is key! Keep working through examples, and you'll become a pro at proving inequalities in no time. Now, go out there and ace those math problems!

By breaking down these proofs step-by-step, we've not only arrived at the correct answers but also strengthened our understanding of essential algebraic techniques. The principles applied can be extended to other areas of algebra and beyond. Practicing such problems will surely sharpen your problem-solving skills. Keep up the excellent work, and you are on the right path to mastering these concepts. Keep practicing, and you'll be well on your way to algebra mastery!