Comparing Numbers Using Greater Than And Less Than Symbols
Comparing numbers is a fundamental skill in mathematics, and understanding how to use the greater than (>) and less than (<) symbols is crucial for grasping inequalities. This article will delve into comparing various types of numbers, including fractions, decimals, and absolute values, using these symbols. We will explore the concepts behind these comparisons and provide clear explanations to help you master this essential mathematical skill. Before diving into the specifics, let's clarify what the symbols > and < actually represent. The greater than symbol (>) indicates that the number on the left is larger than the number on the right. For example, 5 > 3 means that 5 is greater than 3. Conversely, the less than symbol (<) indicates that the number on the left is smaller than the number on the right. For instance, 2 < 7 means that 2 is less than 7. These symbols are the foundation for expressing inequalities and understanding the relative sizes of numbers. In this comprehensive guide, we will walk through several examples, breaking down the steps involved in comparing different types of numbers. From simple fractions to more complex absolute values, you'll gain the confidence to tackle any comparison question. We'll start with a review of basic number comparisons, then move on to fractions, decimals, and finally, absolute values. Each section will build upon the previous one, providing a clear and progressive understanding of the concepts. Whether you're a student looking to improve your math skills or simply someone who wants to brush up on the basics, this article will provide you with the tools and knowledge you need. So, let's begin our journey into the world of number comparisons and master the use of the greater than and less than symbols. By the end of this article, you'll be able to confidently compare any two numbers and express their relationship using the correct inequality symbol. Remember, practice is key to mastering any mathematical skill, so be sure to work through the examples and try some on your own. Let's get started and unlock the secrets of number comparisons!
1) Comparing -3/5 and -3/5
In this first example, we are comparing two identical fractions: -3/5 and -3/5. When comparing two numbers that are exactly the same, the appropriate symbol to use is the equal sign (=). This indicates that the value on the left is identical to the value on the right. Therefore, in this case, -3/5 is equal to -3/5. It's a straightforward comparison, but it sets the stage for understanding more complex comparisons. Understanding equality is the foundation for understanding inequalities. When two numbers are equal, they occupy the same position on the number line. This means there is no difference in their value. This concept becomes particularly important when dealing with more complex expressions and equations. For instance, when solving an equation, the goal is often to find the value of a variable that makes the two sides of the equation equal. Recognizing equality also helps in simplifying expressions. If you encounter identical terms on both sides of an equation or inequality, you can often simplify the expression by canceling them out. This can make the problem easier to solve. In the context of comparing numbers, understanding equality allows us to establish a baseline. Before we can determine if one number is greater than or less than another, we must first recognize when two numbers are the same. This understanding is crucial for building a solid foundation in mathematical comparisons. Moreover, equality plays a crucial role in various mathematical concepts, including geometry, calculus, and statistics. In geometry, for example, two shapes are considered congruent if they are exactly the same, meaning they have the same size and shape. This is a direct application of the concept of equality. In calculus, the concept of limits often involves determining when two expressions become arbitrarily close to each other, which is another way of expressing equality in a limiting sense. In statistics, comparing data sets often involves identifying similarities and differences, and equality is the starting point for these comparisons. Therefore, mastering the concept of equality is not just about recognizing when two numbers are the same; it's about building a fundamental understanding that extends across various areas of mathematics. By grasping this concept, you'll be better equipped to tackle more complex problems and appreciate the interconnectedness of mathematical ideas. Remember, equality is the foundation upon which many other mathematical concepts are built, so it's essential to have a solid understanding of what it means for two numbers or expressions to be equal.
2) Comparing |-2.3| and -2.8
Here, we need to compare the absolute value of -2.3 with -2.8. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, |-2.3| is equal to 2.3. Now we compare 2.3 with -2.8. Since positive numbers are always greater than negative numbers, 2.3 is greater than -2.8. Thus, |-2.3| > -2.8. Understanding absolute value is crucial in this comparison. Absolute value essentially strips away the negative sign from a number, giving us its magnitude or distance from zero. This concept is vital in various mathematical and real-world applications. For example, in physics, absolute value is used to calculate the magnitude of a vector, which represents the length of the vector regardless of its direction. In computer science, absolute value can be used to determine the difference between two values without considering their order. When comparing numbers, absolute value helps us focus on the size of the number rather than its sign. This is particularly useful when dealing with negative numbers, as it allows us to easily compare their magnitudes. In this case, |-2.3| gives us 2.3, which is easier to compare with -2.8. Recognizing that a positive number is always greater than a negative number simplifies the comparison process. Once we have the absolute value, we can directly compare it with the other number. In this example, 2.3 is clearly greater than -2.8, so we can confidently conclude that |-2.3| > -2.8. Moreover, absolute value is a fundamental concept in algebra and calculus. It is used in solving equations and inequalities, as well as in defining continuity and differentiability of functions. Understanding absolute value allows us to work with functions that involve piecewise definitions, where the behavior of the function changes depending on the sign of the input. Absolute value also plays a significant role in the study of limits and sequences. When determining the convergence of a sequence, we often use absolute value to measure the distance between terms and the limit. This helps us understand how the terms of the sequence behave as they approach the limit. Therefore, mastering the concept of absolute value is essential for building a strong foundation in mathematics. It is not just a tool for comparing numbers; it is a fundamental concept that appears in various branches of mathematics and has numerous real-world applications. By understanding absolute value, you'll be better equipped to tackle a wide range of mathematical problems and appreciate its significance in different contexts. Remember, absolute value provides a way to measure the magnitude of a number, which is crucial for comparisons and various mathematical operations.
3) Comparing -0.7 and -0.65
In this scenario, we are comparing two negative decimal numbers: -0.7 and -0.65. When comparing negative numbers, it's important to remember that the number with the smaller absolute value is actually the larger number. This is because it is closer to zero on the number line. To compare these numbers, we can think of -0.7 as -0.70. Now we can easily see that -0.65 is greater than -0.70 because 0.65 is less than 0.70. Therefore, -0.7 < -0.65. Comparing negative numbers can be counterintuitive at first, but understanding the number line makes it much clearer. The number line is a visual representation of numbers, where numbers increase as you move from left to right. Negative numbers are located to the left of zero, and their values decrease as you move further away from zero. This means that -1 is less than 0, -2 is less than -1, and so on. When comparing two negative numbers, the one that is closer to zero is the larger number. This is because it is less negative. In this case, -0.65 is closer to zero than -0.7, so it is the larger number. Visualizing the numbers on a number line can help solidify this concept. Imagine a number line with zero in the middle, -0.65 to the left of zero, and -0.7 further to the left. It becomes clear that -0.65 is closer to zero and therefore greater than -0.7. Another way to think about comparing negative numbers is to imagine them as debts. If you owe $0.70, that's worse than owing $0.65. So, -0.7 is a smaller amount than -0.65. This analogy can help make the comparison more relatable and easier to understand. Understanding how to compare negative numbers is essential for various mathematical applications. It is used in solving inequalities, graphing functions, and analyzing data sets. For example, when working with temperature scales, negative numbers represent temperatures below zero. Comparing these temperatures involves the same principles as comparing any negative numbers. Moreover, the concept of comparing negative numbers extends beyond decimal numbers. It applies to fractions, integers, and any other type of number. The key is to remember that the number with the smaller absolute value is the larger number when dealing with negative numbers. Therefore, mastering the comparison of negative numbers is a crucial skill for building a strong foundation in mathematics. It is a concept that appears in various contexts and is essential for understanding more advanced topics. By visualizing the number line and understanding the relationship between negative numbers and zero, you can confidently compare any two negative numbers and determine which one is greater or less than the other.
4) Comparing 3/4 and |-0.8|
In this comparison, we have a fraction, 3/4, and the absolute value of a decimal, |-0.8|. First, we need to find the absolute value of -0.8, which is 0.8. Next, we need to compare 3/4 and 0.8. To make the comparison easier, we can convert 3/4 to a decimal. 3/4 is equal to 0.75. Now we compare 0.75 and 0.8. Since 0.75 is less than 0.8, we can say that 3/4 < |-0.8|. Converting between fractions and decimals is a fundamental skill in mathematics, especially when comparing numbers. Fractions and decimals are simply different ways of representing the same value, and being able to switch between them allows for easier comparisons and calculations. In this case, we had to compare a fraction (3/4) with the absolute value of a decimal (|-0.8|). To make the comparison straightforward, we converted both numbers to the same format, which in this case was decimals. Converting a fraction to a decimal involves dividing the numerator (the top number) by the denominator (the bottom number). So, 3/4 becomes 3 ÷ 4, which equals 0.75. This conversion allows us to directly compare the value of 3/4 with the decimal 0.8. Alternatively, we could have converted the decimal 0.8 back into a fraction. 0.8 is equivalent to 8/10, which can be simplified to 4/5. Now we would be comparing 3/4 and 4/5. To compare these fractions, we would need to find a common denominator. The least common multiple of 4 and 5 is 20. So, we would convert 3/4 to 15/20 and 4/5 to 16/20. Now it's clear that 15/20 (3/4) is less than 16/20 (4/5 or 0.8). Choosing the best method for conversion depends on the specific numbers being compared and personal preference. Sometimes converting to decimals is easier, while other times converting to fractions with a common denominator is more convenient. The key is to be comfortable with both methods and choose the one that makes the comparison clearest. Moreover, the ability to convert between fractions and decimals is crucial in various mathematical contexts. It is used in solving equations, working with percentages, and understanding ratios and proportions. In practical applications, such as cooking or measuring, you often need to convert between fractions and decimals to accurately follow recipes or make precise measurements. Therefore, mastering this skill is not just about comparing numbers; it's about building a versatile mathematical tool that can be applied in a wide range of situations. By practicing converting between fractions and decimals, you'll become more confident in your ability to compare numbers and solve mathematical problems efficiently.
5) Comparing 1/8 and 1/9
Here, we are comparing two fractions, 1/8 and 1/9. When comparing fractions with the same numerator (the top number), the fraction with the smaller denominator (the bottom number) is the larger fraction. This is because the whole is being divided into fewer parts, so each part is larger. In this case, 1/8 is greater than 1/9 because 8 is less than 9. Thus, 1/8 > 1/9. Understanding fractions and how their denominators affect their value is crucial for this comparison. A fraction represents a part of a whole, with the numerator indicating how many parts we have and the denominator indicating how many parts the whole is divided into. When the numerator is the same, as in this case, the denominator plays a key role in determining the size of the fraction. A smaller denominator means that the whole is divided into fewer parts, so each part is larger. Conversely, a larger denominator means that the whole is divided into more parts, so each part is smaller. Imagine a pie being divided into slices. If you cut the pie into 8 slices, each slice will be larger than if you cut the same pie into 9 slices. This is why 1/8 is greater than 1/9. This concept is fundamental to comparing fractions and understanding their relative sizes. It is also important for performing operations with fractions, such as addition and subtraction, where finding a common denominator is essential. The ability to compare fractions is not only important in mathematics but also in everyday life. For example, when sharing a pizza, you might need to compare fractions to determine who gets the larger slice. Similarly, when following a recipe, you might need to adjust the quantities of ingredients based on fractional amounts. Understanding how denominators affect fraction size also helps in estimating and approximating values. If you know that 1/8 is slightly larger than 1/9, you can use this knowledge to make quick estimations in various situations. Moreover, this principle extends to more complex comparisons involving multiple fractions. By understanding the relationship between the denominator and the fraction's value, you can easily rank fractions with the same numerator from largest to smallest. Therefore, mastering the comparison of fractions with the same numerator is a crucial step in building a strong foundation in fraction operations. It is a concept that appears in various mathematical contexts and has numerous practical applications. By visualizing the parts of a whole and understanding how the denominator affects their size, you can confidently compare fractions and make accurate judgments about their values.
6) Comparing -1 3/4 and -1.75
In this example, we are comparing a mixed number, -1 3/4, and a decimal, -1.75. To compare them, we need to convert the mixed number to a decimal or the decimal to a fraction. Converting -1 3/4 to a decimal, we first recognize that 3/4 is equal to 0.75. Therefore, -1 3/4 is equal to -1.75. Now we are comparing -1.75 and -1.75. Since they are the same, -1 3/4 = -1.75. Converting between mixed numbers and decimals is an essential skill for comparing and performing operations with different types of numbers. A mixed number is a combination of a whole number and a fraction, while a decimal is a number expressed in base-10 notation, with digits to the right of a decimal point representing fractional parts. Being able to convert between these forms allows us to compare numbers more easily and perform calculations more efficiently. In this case, we had to compare the mixed number -1 3/4 with the decimal -1.75. To make the comparison straightforward, we converted the mixed number to a decimal. To convert a mixed number to a decimal, we first focus on the fractional part. In this case, the fractional part is 3/4. We convert this fraction to a decimal by dividing the numerator (3) by the denominator (4), which gives us 0.75. Then, we combine this decimal with the whole number part of the mixed number. In this case, the whole number part is -1, so we have -1 + (-0.75), which equals -1.75. Now we can easily compare -1.75 and -1.75, and we see that they are equal. Alternatively, we could have converted the decimal -1.75 to a mixed number. The whole number part is -1. The decimal part, 0.75, can be expressed as the fraction 75/100. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 25. So, 75/100 simplifies to 3/4. Therefore, -1.75 is equivalent to -1 3/4. Choosing the best method for conversion depends on the specific numbers being compared and personal preference. Sometimes converting to decimals is easier, while other times converting to fractions or mixed numbers is more convenient. The key is to be comfortable with both methods and choose the one that makes the comparison clearest. Moreover, the ability to convert between mixed numbers and decimals is crucial in various mathematical contexts. It is used in solving equations, working with measurements, and understanding proportions. In practical applications, such as cooking or construction, you often need to convert between mixed numbers and decimals to accurately follow instructions or make precise measurements. Therefore, mastering this skill is not just about comparing numbers; it's about building a versatile mathematical tool that can be applied in a wide range of situations. By practicing converting between mixed numbers and decimals, you'll become more confident in your ability to compare numbers and solve mathematical problems efficiently.
7) Comparing -5/2 and -3
In this scenario, we are comparing a fraction, -5/2, and an integer, -3. To compare these numbers, we need to convert the fraction to an integer or the integer to a fraction with the same denominator. Let's convert -5/2 to a decimal first. -5 divided by 2 is -2.5. Now we compare -2.5 and -3. Remember that with negative numbers, the number closer to zero is larger. -2.5 is closer to zero than -3, so -2.5 is greater than -3. Therefore, -5/2 > -3. Alternatively, we can convert -3 to a fraction with a denominator of 2. -3 is equal to -6/2. Now we compare -5/2 and -6/2. Since -5 is greater than -6, -5/2 is greater than -6/2. This confirms that -5/2 > -3. Comparing fractions and integers often involves converting them to a common format, either both as fractions or both as decimals. This makes the comparison much more straightforward. In this case, we had the fraction -5/2 and the integer -3. We demonstrated two methods for comparing them: converting the fraction to a decimal and converting the integer to a fraction with the same denominator. Converting -5/2 to a decimal involves dividing -5 by 2, which results in -2.5. Now we can easily compare -2.5 and -3. Since negative numbers become smaller as their absolute value increases, -2.5 is greater than -3 because it is closer to zero. The other method involves converting -3 to a fraction with a denominator of 2. To do this, we multiply -3 by 2/2, which gives us -6/2. Now we can compare -5/2 and -6/2. Since they have the same denominator, we simply compare the numerators. -5 is greater than -6, so -5/2 is greater than -6/2. This method reinforces the concept of comparing fractions with common denominators. When fractions have the same denominator, the fraction with the larger numerator is the larger fraction. This is because both fractions represent parts of the same whole, and the numerator indicates how many of those parts are being considered. Moreover, the ability to compare fractions and integers is crucial in various mathematical contexts. It is used in solving inequalities, graphing functions, and analyzing data sets. For example, when working with number lines, you need to be able to accurately place both fractions and integers to understand their relative positions. In practical applications, comparing fractions and integers is important for making informed decisions. For example, if you are comparing the prices of two items, one expressed as a fraction and the other as an integer, you need to be able to convert them to a common format to determine which is cheaper. Therefore, mastering the comparison of fractions and integers is a crucial skill for building a strong foundation in mathematics. It is a concept that appears in various contexts and is essential for understanding more advanced topics. By practicing converting between fractions and integers and understanding the principles of comparing negative numbers, you can confidently compare any fraction and integer and determine their relative values.
8) Comparing |-0.6| and |-0.55|
In this example, we are comparing the absolute values of two decimal numbers: |-0.6| and |-0.55|. First, we need to find the absolute value of each number. The absolute value of -0.6 is 0.6, and the absolute value of -0.55 is 0.55. Now we compare 0.6 and 0.55. Since 0.6 is greater than 0.55, we can say that |-0.6| > |-0.55|. Understanding absolute value is essential for this comparison. As we discussed earlier, the absolute value of a number is its distance from zero on the number line, regardless of direction. This means that the absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart. In this case, we are dealing with the absolute values of two negative decimal numbers. The absolute value of -0.6 is 0.6, and the absolute value of -0.55 is 0.55. Once we have found the absolute values, the comparison becomes straightforward. We are simply comparing two positive decimal numbers. To compare 0.6 and 0.55, we can think of 0.6 as 0.60. Now it's clear that 0.60 is greater than 0.55. Therefore, we can conclude that |-0.6| > |-0.55|. This example highlights the importance of understanding how absolute value affects the comparison of numbers. By stripping away the negative signs, we focus on the magnitudes of the numbers and compare their distances from zero. This is particularly useful when dealing with situations where the direction or sign of a number is not relevant, and we are only interested in its size. Moreover, absolute value is a fundamental concept in algebra and calculus. It is used in solving equations and inequalities, as well as in defining continuity and differentiability of functions. Understanding absolute value allows us to work with functions that involve piecewise definitions, where the behavior of the function changes depending on the sign of the input. Absolute value also plays a significant role in the study of limits and sequences. When determining the convergence of a sequence, we often use absolute value to measure the distance between terms and the limit. This helps us understand how the terms of the sequence behave as they approach the limit. Therefore, mastering the concept of absolute value is essential for building a strong foundation in mathematics. It is not just a tool for comparing numbers; it is a fundamental concept that appears in various branches of mathematics and has numerous real-world applications. By understanding absolute value, you'll be better equipped to tackle a wide range of mathematical problems and appreciate its significance in different contexts.
9) Comparing 3/4 and |-3 3/5|
In this case, we are comparing a fraction, 3/4, with the absolute value of a mixed number, |-3 3/5|. First, we need to find the absolute value of -3 3/5. The absolute value of -3 3/5 is 3 3/5. Next, we need to compare 3/4 and 3 3/5. It's clear that 3 3/5 is greater than 3/4 because 3 3/5 is greater than 1, while 3/4 is less than 1. Thus, 3/4 < |-3 3/5|. Comparing fractions and mixed numbers, especially when one involves an absolute value, requires careful attention to the values represented. In this example, we have 3/4 on one side and the absolute value of -3 3/5 on the other side. The first step is to find the absolute value of -3 3/5. The absolute value of a number is its distance from zero, so the absolute value of -3 3/5 is simply 3 3/5. Now we are comparing 3/4 and 3 3/5. At this point, we can easily see that 3 3/5 is much larger than 3/4. 3/4 is a fraction less than 1, while 3 3/5 is a mixed number greater than 3. Therefore, there is no need for further conversion or calculation; we can immediately conclude that 3/4 is less than 3 3/5. This example illustrates the importance of developing a sense of number magnitude. By recognizing the relative sizes of different types of numbers, we can often make comparisons quickly and efficiently. In this case, knowing that 3/4 is less than 1 and 3 3/5 is greater than 3 allows us to bypass more complex calculations and directly determine the inequality. However, if we wanted to be more precise or if the numbers were closer in value, we could convert both numbers to fractions or decimals. To convert 3 3/5 to an improper fraction, we multiply the whole number (3) by the denominator (5) and add the numerator (3), giving us 18. So, 3 3/5 is equal to 18/5. Now we can compare 3/4 and 18/5. To compare these fractions, we need a common denominator. The least common multiple of 4 and 5 is 20. So, we convert 3/4 to 15/20 and 18/5 to 72/20. Now it's clear that 15/20 is less than 72/20, confirming our earlier conclusion. Therefore, mastering the comparison of fractions and mixed numbers involves both understanding the relative sizes of numbers and being able to perform conversions when necessary. By developing these skills, you'll be able to confidently compare any two numbers and make accurate judgments about their values.
10) Comparing -3/4 and -3/5
Here, we are comparing two negative fractions, -3/4 and -3/5. When comparing negative fractions, it's helpful to remember that the fraction with the smaller absolute value is the larger number. To compare these fractions, we can find a common denominator. The least common multiple of 4 and 5 is 20. So, we convert -3/4 to -15/20 and -3/5 to -12/20. Now we compare -15/20 and -12/20. Since -12 is greater than -15, -12/20 is greater than -15/20. Therefore, -3/4 < -3/5. Comparing negative fractions requires a careful understanding of both fraction values and the concept of negative numbers. As we discussed earlier, negative numbers become smaller as their absolute value increases. This means that when comparing two negative fractions, the one with the smaller absolute value is actually the larger number. In this case, we are comparing -3/4 and -3/5. The first step is to find a common denominator. The least common multiple of 4 and 5 is 20, so we will convert both fractions to have a denominator of 20. To convert -3/4 to a fraction with a denominator of 20, we multiply both the numerator and the denominator by 5: -3/4 * (5/5) = -15/20. To convert -3/5 to a fraction with a denominator of 20, we multiply both the numerator and the denominator by 4: -3/5 * (4/4) = -12/20. Now we can compare -15/20 and -12/20. Since they have the same denominator, we simply compare the numerators. Remember that we are dealing with negative numbers, so the number with the smaller absolute value is the larger number. -12 is greater than -15, so -12/20 is greater than -15/20. Therefore, we can conclude that -3/4 < -3/5. Alternatively, we could have visualized these fractions on a number line. A number line helps to illustrate the relative positions of numbers and makes it easier to compare them. Imagine a number line with zero in the middle, negative numbers to the left, and positive numbers to the right. -3/4 is located to the left of zero, and -3/5 is also located to the left of zero, but closer to zero than -3/4. This visual representation makes it clear that -3/5 is greater than -3/4. Moreover, the ability to compare negative fractions is crucial in various mathematical contexts. It is used in solving inequalities, graphing functions, and analyzing data sets. For example, when working with financial data, you might need to compare negative fractions to determine which investment had a smaller loss. Therefore, mastering the comparison of negative fractions is a crucial skill for building a strong foundation in mathematics. It is a concept that appears in various contexts and is essential for understanding more advanced topics. By practicing converting fractions to a common denominator and understanding the principles of comparing negative numbers, you can confidently compare any two negative fractions and determine their relative values.
11) Comparing |-4 1/2| and 9/2
In this scenario, we are comparing the absolute value of a mixed number, |-4 1/2|, with a fraction, 9/2. First, we need to find the absolute value of -4 1/2, which is 4 1/2. Next, we need to compare 4 1/2 and 9/2. To make this comparison easier, we can convert the mixed number to an improper fraction. 4 1/2 is equal to (4 * 2 + 1) / 2 = 9/2. Now we are comparing 9/2 and 9/2. Since they are the same, |-4 1/2| = 9/2. Converting mixed numbers to improper fractions and understanding absolute values are key to solving this comparison. As we've discussed, the absolute value of a number is its distance from zero, so |-4 1/2| is simply 4 1/2. Now we need to compare 4 1/2 and 9/2. To compare a mixed number and a fraction, it's often easiest to convert the mixed number to an improper fraction. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). To convert 4 1/2 to an improper fraction, we multiply the whole number (4) by the denominator (2) and add the numerator (1): (4 * 2) + 1 = 9. This becomes the new numerator, and we keep the same denominator, so 4 1/2 is equal to 9/2. Now we are comparing 9/2 and 9/2. Since they are the same, we can confidently say that |-4 1/2| = 9/2. This example demonstrates a common strategy in mathematics: converting numbers to a common format to make comparisons easier. By converting the mixed number to an improper fraction, we were able to directly compare it with the other fraction. Moreover, this example reinforces the importance of understanding absolute value. By taking the absolute value of -4 1/2, we removed the negative sign and focused on the magnitude of the number. This allowed us to compare the distances from zero, which is the essence of absolute value. Therefore, mastering the conversion of mixed numbers to improper fractions and understanding absolute values are crucial skills for building a strong foundation in number comparisons. These skills are used in various mathematical contexts and are essential for understanding more advanced topics.
12) Comparing 7/4 and 3/2
In this final comparison, we are comparing two fractions, 7/4 and 3/2. To compare these fractions, we need to find a common denominator. The least common multiple of 4 and 2 is 4. So, we convert 3/2 to a fraction with a denominator of 4. To do this, we multiply both the numerator and the denominator by 2: 3/2 * (2/2) = 6/4. Now we compare 7/4 and 6/4. Since 7 is greater than 6, 7/4 is greater than 6/4. Therefore, 7/4 > 3/2. Finding a common denominator is the key to comparing fractions with different denominators. A common denominator allows us to express the fractions in terms of the same-sized parts, making the comparison straightforward. In this case, we are comparing 7/4 and 3/2. The first step is to find the least common multiple (LCM) of the denominators, which are 4 and 2. The LCM of 4 and 2 is 4, so we will convert both fractions to have a denominator of 4. 7/4 already has a denominator of 4, so we don't need to change it. To convert 3/2 to a fraction with a denominator of 4, we need to multiply both the numerator and the denominator by the same number. In this case, we multiply by 2: 3/2 * (2/2) = 6/4. Now we are comparing 7/4 and 6/4. Since they have the same denominator, we simply compare the numerators. 7 is greater than 6, so 7/4 is greater than 6/4. Therefore, we can conclude that 7/4 > 3/2. This example illustrates the importance of understanding how to find a common denominator and how it simplifies fraction comparisons. By expressing fractions in terms of the same-sized parts, we can directly compare their values and determine which one is larger or smaller. Moreover, the ability to compare fractions is crucial in various mathematical contexts. It is used in solving equations, working with ratios and proportions, and understanding probability. In practical applications, comparing fractions is important for making informed decisions. For example, if you are comparing the sizes of two pizzas, you need to be able to compare fractions to determine which pizza is larger. Therefore, mastering the comparison of fractions is a crucial skill for building a strong foundation in mathematics. It is a concept that appears in various contexts and is essential for understanding more advanced topics. By practicing finding common denominators and comparing numerators, you can confidently compare any two fractions and determine their relative values.
Conclusion
In conclusion, using the symbols > and < to compare numbers is a fundamental skill in mathematics. We've explored various scenarios, including comparing fractions, decimals, absolute values, and negative numbers. By understanding the concepts and practicing the techniques discussed in this article, you can confidently compare any two numbers and express their relationship using the correct inequality symbol. Remember, the greater than symbol (>) indicates that the number on the left is larger than the number on the right, while the less than symbol (<) indicates that the number on the left is smaller than the number on the right. Keep practicing, and you'll master this essential skill in no time!
