Solving |2p + 4| < 3 Finding The Smallest Integer P
Introduction
In this article, we will delve into the world of inequalities and absolute values. Our main focus will be on solving the inequality |2p + 4| < 3 and pinpointing the smallest integer value of p that satisfies this condition. This type of problem is a cornerstone of basic algebra and is frequently encountered in various mathematical contexts. Understanding how to tackle such inequalities is not just crucial for academic success but also for developing logical reasoning and problem-solving skills that are applicable in many real-world scenarios.
The concept of absolute value can sometimes seem tricky, but it's a fundamental tool in mathematics. Remember, the absolute value of a number is its distance from zero, regardless of direction. This means that |x| represents both x and -x. When dealing with inequalities involving absolute values, we need to consider both the positive and negative cases to ensure we capture all possible solutions. The key here is to break down the absolute value inequality into two separate inequalities, each representing a different scenario. This allows us to work with more straightforward algebraic expressions and find the range of values that satisfy the original condition.
By the end of this discussion, you'll not only know the answer to the specific problem but also have a stronger grasp of the underlying principles. We'll break down each step in detail, providing explanations and insights along the way. Whether you're a student grappling with algebra, a math enthusiast looking to refresh your knowledge, or simply someone who enjoys a good mathematical puzzle, this article is designed to help you understand and appreciate the elegance of solving inequalities.
Understanding Absolute Value Inequalities
Before we dive into the specifics of our problem, let's solidify our understanding of absolute value inequalities. As mentioned earlier, the absolute value of a number is its distance from zero. This concept is crucial when dealing with inequalities because it introduces two possibilities: the expression inside the absolute value can be either positive or negative, but its distance from zero must still be less than the given bound. In our case, the inequality |2p + 4| < 3 tells us that the expression 2p + 4 must be within 3 units of zero, meaning it can be between -3 and 3.
To solve an absolute value inequality of the form |ax + b| < c, we need to consider two separate inequalities:
- ax + b < c
- -(ax + b) < c, which is equivalent to ax + b > -c
These two inequalities define the range of values for x that satisfy the original absolute value inequality. Graphically, this represents the region on the number line where the distance of ax + b from zero is less than c. It's essential to understand why we split the absolute value inequality into two cases. The first case considers the situation where the expression inside the absolute value is positive or zero, while the second case considers the situation where it's negative. By addressing both possibilities, we ensure that we capture all solutions.
This approach allows us to transform a single absolute value inequality into a compound inequality, which is often easier to solve. By breaking down the problem into smaller, more manageable parts, we can systematically find the solution set. In the following sections, we'll apply this technique to our specific inequality, |2p + 4| < 3, and walk through the steps to find the smallest integer value of p that satisfies it. Remember, the key is to carefully consider both cases and solve each inequality separately, then combine the results to find the overall solution.
Solving the Inequality |2p + 4| < 3
Now, let's apply our understanding of absolute value inequalities to the problem at hand: |2p + 4| < 3. As we discussed, this inequality implies that the expression 2p + 4 must be within 3 units of zero. To solve this, we'll break it down into two separate inequalities:
- 2p + 4 < 3
- -(2p + 4) < 3, which is equivalent to 2p + 4 > -3
Let's solve the first inequality, 2p + 4 < 3. To isolate p, we first subtract 4 from both sides:
2p < 3 - 4
2p < -1
Next, we divide both sides by 2:
p < -1/2
So, the first inequality tells us that p must be less than -1/2.
Now, let's solve the second inequality, 2p + 4 > -3. Again, we start by subtracting 4 from both sides:
2p > -3 - 4
2p > -7
Then, we divide both sides by 2:
p > -7/2
So, the second inequality tells us that p must be greater than -7/2. Combining these two results, we find that p must satisfy both conditions: p < -1/2 and p > -7/2. This means that p lies between -7/2 and -1/2. In other words, -7/2 < p < -1/2. This range represents all the possible values of p that make the original inequality true.
Identifying the Smallest Integer Solution
Now that we've determined the range of values for p that satisfy the inequality |2p + 4| < 3, our next step is to find the smallest integer within that range. We know that -7/2 < p < -1/2. Let's convert these fractions to decimals to make it easier to visualize the range on the number line. -7/2 is equal to -3.5, and -1/2 is equal to -0.5. So, we're looking for the smallest integer p such that -3.5 < p < -0.5.
Think about the integers on the number line. We have -3, -2, -1, and 0. Which of these fall within our range? -3.5 < p < -0.5. The integer -3 is greater than -3.5, and -1 is less than -0.5.
The integers that fall within this range are -3, -2, and -1. Remember, we're looking for the smallest integer. Comparing these values, we can see that -3 is the smallest integer that satisfies the inequality. Therefore, the smallest integer value of p that makes the inequality |2p + 4| < 3 true is -3. This solution not only answers the specific question but also reinforces our understanding of how to work with absolute value inequalities and identify solutions within a given range.
Conclusion
In conclusion, we successfully solved the inequality |2p + 4| < 3 and found that the smallest integer value of p that satisfies it is -3. We achieved this by first understanding the concept of absolute value inequalities and how they can be broken down into two separate cases. By solving these individual inequalities and combining their results, we determined the range of possible values for p. Finally, we identified the smallest integer within that range.
This exercise highlights the importance of a systematic approach to solving mathematical problems. By breaking down complex problems into simpler steps, we can make them more manageable and increase our chances of finding the correct solution. Understanding the underlying principles, such as the definition of absolute value and the properties of inequalities, is crucial for success in algebra and beyond. The ability to solve inequalities is not just a mathematical skill; it's a valuable tool for logical reasoning and problem-solving in various contexts.
The process of solving |2p + 4| < 3 exemplifies a common strategy in mathematics: transforming a problem into an equivalent form that is easier to solve. By splitting the absolute value inequality into two linear inequalities, we were able to apply familiar algebraic techniques to isolate the variable p. This approach can be applied to a wide range of problems involving absolute values and other mathematical concepts. We hope this exploration has not only provided a clear solution to the problem but also deepened your understanding of the principles involved. Remember, practice and persistence are key to mastering mathematical concepts and building confidence in your problem-solving abilities.
