Problem 51 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<\) or \(>\). If \(3

Step-by-Step Solution

Answer

The correct symbol to use is \(>\), so the completed statement is \(y > 3\).

Step 1: Analyzing the given condition

The problem mentions that \(3

Step 2: Applying the condition to complete the statement

Using the condition, 'y is greater than 3', we can complete the statement as 'y > 3'.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (such as x, y, or z), and operation symbols (like +, −, ×, ÷). For instance, the expression \(3 < y\) from the exercise contains a number (3), a variable (y), and an inequality symbol (<), which tells us about the relationship between the number and the variable. Understanding how to construct and interpret these expressions is essential in algebra, as they are used to represent real-world situations and solve problems.

In our exercise, by stating \(3 < y\), we effectively create an algebraic expression that compares a known value to an unknown one, which is the variable. This opens up the concept of variables being placeholders for any number that satisfies the given condition. In algebra, much of your success will hinge on how well you can manipulate these expressions to find solutions to equations and inequalities.

Comparing Numbers

Comparing numbers is one of the foundational skills in mathematics. It involves determining the relative size of two numerical values. When we compare numbers, we assess whether they are equal to, less than, or greater than each other. In the context of the exercise, we are asked to compare the variable y with the number 3. The problem states \(3 < y\), telling us immediately that 3 is smaller than whatever value y holds.

This skill goes beyond simply looking at the numbers; it extends to understanding their numerical value in different contexts, such as with negative numbers, fractions, and even in terms of variables, where the exact values may not be known. In real-world applications, comparing numbers can be extremely useful for making decisions based on data and for solving problems where number relations play a critical role.

Inequality Symbols

Inequality symbols are the shorthand we use in mathematics to show the relationship between two values when they are not equal. The most common symbols are \(>\) for 'greater than' and \(<\) for 'less than'. There are also symbols to represent 'greater than or equal to' (\(\geq\)) and 'less than or equal to' (\(\leq\)). In our exercise, the symbol used is \(<\), which tells us that the number on the left side of the symbol (3) is smaller than the number (or in this case, the variable) on the right side (y).

These symbols are not just simple annotations; they denote the relative positions of numbers on the number line and are key to understanding the scope of possible solutions to inequalities in algebra. When using inequality symbols, it's critical to remember that they point towards the smaller of the two numbers, effectively 'eating' the larger number—this visual can aid memory and understanding of these symbols.

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Problem 51

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Problem 52

Other exercises in this chapter

Problem 51

Find the sum. Use a calculator if you wish. $$10.97+(-51.14)+(-40.97)$$

View solution

Problem 51

You rappel down the side of a mountain at a rate of 2 feet per second. What is your change in position after rappelling for 10 seconds?

View solution

Problem 52

Find the terms of the expression. $$ -10+4 b $$

View solution

Problem 52

Find the domain of the function. $$ y=\frac{10-x}{7-x} $$

View solution

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (such as x, y, or z), and operation symbols (like +, −, ×, ÷). For instance, the expression \(3 < y\) from the exercise contains a number (3), a variable (y), and an inequality symbol (<), which tells us about the relationship between the number and the variable. Understanding how to construct and interpret these expressions is essential in algebra, as they are used to represent real-world situations and solve problems.

In our exercise, by stating \(3 < y\), we effectively create an algebraic expression that compares a known value to an unknown one, which is the variable. This opens up the concept of variables being placeholders for any number that satisfies the given condition. In algebra, much of your success will hinge on how well you can manipulate these expressions to find solutions to equations and inequalities.

Comparing Numbers

Comparing numbers is one of the foundational skills in mathematics. It involves determining the relative size of two numerical values. When we compare numbers, we assess whether they are equal to, less than, or greater than each other. In the context of the exercise, we are asked to compare the variable y with the number 3. The problem states \(3 < y\), telling us immediately that 3 is smaller than whatever value y holds.

This skill goes beyond simply looking at the numbers; it extends to understanding their numerical value in different contexts, such as with negative numbers, fractions, and even in terms of variables, where the exact values may not be known. In real-world applications, comparing numbers can be extremely useful for making decisions based on data and for solving problems where number relations play a critical role.

Inequality Symbols

Inequality symbols are the shorthand we use in mathematics to show the relationship between two values when they are not equal. The most common symbols are \(>\) for 'greater than' and \(<\) for 'less than'. There are also symbols to represent 'greater than or equal to' (\(\geq\)) and 'less than or equal to' (\(\leq\)). In our exercise, the symbol used is \(<\), which tells us that the number on the left side of the symbol (3) is smaller than the number (or in this case, the variable) on the right side (y).

These symbols are not just simple annotations; they denote the relative positions of numbers on the number line and are key to understanding the scope of possible solutions to inequalities in algebra. When using inequality symbols, it's critical to remember that they point towards the smaller of the two numbers, effectively 'eating' the larger number—this visual can aid memory and understanding of these symbols.