Mastering Inequalities: Step-by-Step Justification
When you're tackling an inequality, it's not just about getting the right answer; it's about understanding why each step you take is valid. Think of it like building a house – each brick needs to be placed correctly, and you need to know why it goes there to ensure the whole structure is sound. Today, we're going to break down the inequality $3 x+ rac{5}{8} \geq 4 x- rac{1}{2}$ and justify every single move we make. This process helps solidify your understanding of algebraic manipulation and the fundamental properties of inequalities. Let's dive in!
The Starting Point: Understanding the Inequality
Our journey begins with the inequality $3 x+ rac{5}{8} \geq 4 x- rac{1}{2}$. This statement is a comparison, telling us that the expression on the left side ($3x + \frac{5}{8}$) is greater than or equal to the expression on the right side ($4x - \frac{1}{2}$). The goal is to isolate the variable 'x' to find the range of values for 'x' that make this statement true. Just like solving an equation, we can perform operations on both sides of the inequality, but we must be mindful of how these operations affect the inequality sign. The core principle is to maintain the balance of the comparison. We're not just moving numbers around randomly; each action is governed by specific mathematical properties that preserve the truth of the original statement. Understanding this foundational concept is crucial before we even make our first move. It’s about maintaining the integrity of the mathematical relationship between the two expressions. Remember, inequalities describe a set of numbers, not just a single value, and our steps must lead us to accurately define that set.
Step 1: Clearing the Fractions
One of the first things you might notice are the fractions: $ rac{5}{8}$ and $ rac{1}{2}$. While we can work with fractions, often it's easier to eliminate them by multiplying both sides of the inequality by a common denominator. The least common multiple (LCM) of 8 and 2 is 8. So, we'll multiply every term on both sides of the inequality by 8. Why is this justified? It's a direct application of the Multiplication Property of Inequality. This property states that if you multiply both sides of an inequality by a positive number, the direction of the inequality sign remains the same. Since 8 is positive, our 'greater than or equal to' sign ( \geq) stays put. Let's see how this plays out:
8 \times \left(3 x+ rac{5}{8}\right) \geq 8 \times \left(4 x- rac{1}{2}\right)
Now, distribute the 8:
(8 \times 3x) + \left(8 \times rac{5}{8}\right) \geq \left(8 \times 4x\right) - \left(8 \times rac{1}{2}\right)
Simplifying:
See? No more fractions! This step, justified by the Multiplication Property of Inequality (multiplying by a positive number), has made our inequality much cleaner and easier to work with. It's a strategic move to simplify the problem without altering the solution set. This is a key concept in algebraic problem-solving: simplify first, then solve.
Step 2: Consolidating Variable Terms
Now that our fractions are gone, we want to gather all the terms containing 'x' on one side of the inequality and all the constant terms on the other. Let's choose to move the 'x' terms to the right side. We can do this by subtracting $24x$ from both sides. This action is justified by the Addition Property of Inequality. This property is straightforward: you can add or subtract the same number from both sides of an inequality, and the direction of the inequality sign remains unchanged. Since we are subtracting $24x$, we are essentially adding a negative $24x$, which falls under this property.
On the left side, $24x - 24x$ cancels out, leaving just 5. On the right side, $40x - 24x$ gives us $16x$. Our inequality now looks like this:
This step is crucial because it starts to isolate the variable term. By moving the 'x' terms together, we are reducing the complexity of the inequality and getting closer to finding the value of 'x'. The Addition Property of Inequality is one of the most fundamental tools we use, ensuring that whatever operation we perform on one side to balance the equation, we must perform the exact same operation on the other side to maintain the inequality's truth. It’s like moving weights on a scale – if you take weight off one side, you must take the same amount off the other to keep it balanced.
Step 3: Isolating the Variable Term
We're getting closer! Our inequality is now $5 \geq 16x - 4$. The 'x' term ($16x$) is on the right side, but it's still associated with a '-4'. To isolate the $16x$ term, we need to get rid of that '-4'. We can do this by adding 4 to both sides of the inequality. Again, this is justified by the Addition Property of Inequality. Adding 4 to both sides doesn't change the direction of the inequality sign.
Simplifying both sides:
This is a significant step. We have successfully isolated the term containing our variable. The inequality now clearly shows the relationship between a constant (9) and the variable term ($16x$). This systematic approach, using the Addition Property of Inequality, ensures that we are not making any unwarranted changes to the problem. It’s about systematically stripping away the constants and coefficients that are not directly attached to 'x', one by one, ensuring that the original relationship of greater than or equal to is preserved at every stage. This methodical process is what distinguishes a rigorous mathematical solution from a guess.
Step 4: Solving for 'x'
We're at the home stretch! Our inequality is $9 \geq 16x$. To find the value of 'x', we need to undo the multiplication by 16. We do this by dividing both sides of the inequality by 16. This is justified by the Division Property of Inequality. Similar to the multiplication property, if you divide both sides of an inequality by a positive number, the direction of the inequality sign remains the same. Since 16 is positive, our ' \geq' sign stays as it is.
rac{9}{16} \geq rac{16x}{16}
Simplifying:
rac{9}{16} \geq x
And there you have it! We've successfully solved for 'x'. This final step, dividing by a positive number, is governed by the Division Property of Inequality. It's the culmination of all the previous steps, each justified by fundamental algebraic properties. We have transformed the original complex inequality into a simple statement that tells us the range of values 'x' can take. The solution means that any number less than or equal to $ rac{9}{16}$ will satisfy the original inequality. It’s important to remember that if we had divided by a negative number at this stage, we would have had to reverse the direction of the inequality sign. But since 16 is positive, we're good!
Conclusion: The Power of Justification
As we've seen, solving an inequality isn't just about algebraic manipulation; it's about understanding the 'why' behind each step. From clearing fractions using the Multiplication Property of Inequality to isolating 'x' using the Addition and Division Properties, each action is deliberate and mathematically sound. These properties are the bedrock of algebra, ensuring that our solutions are accurate and reliable. By justifying each step, you not only deepen your understanding of inequalities but also build a stronger foundation for more complex mathematical concepts. Remember, every valid mathematical operation preserves the truth of the original statement, leading us step-by-step to the correct solution set.
For further exploration into the properties of inequalities and how they are applied in various mathematical contexts, I recommend visiting Khan Academy's comprehensive resources on algebra. Their detailed explanations and practice problems can help solidify your understanding. Another excellent source for mathematical learning is Brilliant.org, which offers interactive courses that make learning complex topics engaging and intuitive.
