Problem 80 - Algebra A Combined Function

Question

Write each algebraic expression described. Simplify if possible. See Example \(11 .\) If \(x\) is the first of two consecutive integers, express the sum of 20 and the second consecutive integer as an algebraic expression containing the variable \(x\).

Step-by-Step Solution

Answer

The algebraic expression is \(x + 21\).

Step 1: Understanding Consecutive Integers

Consecutive integers are numbers that follow each other in order. If the first integer is given as \(x\), then the next consecutive integer will be \(x + 1\). This is because each consecutive integer is one more than the previous integer.

Step 2: Expressing the Second Integer

Given that the first integer is \(x\), the second consecutive integer can be expressed as \(x + 1\). We now have expressions for both integers: \(x\) for the first and \(x + 1\) for the second integer.

Step 3: Finding the Sum with 20

The problem asks for the sum of 20 and the second consecutive integer. Thus, we need to add 20 to the second integer: \(20 + (x + 1)\).

Step 4: Simplifying the Expression

Simplify the expression obtained in the previous step: \(20 + (x + 1) = 20 + x + 1\). Further combine the constant terms: \(20 + 1 = 21\), leading to the simplified expression \(x + 21\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators (such as +, −, ×, ÷). In the context of consecutive integers, algebraic expressions help us model real-world problems using variables. For example, when dealing with consecutive integers starting with a number represented by the variable \(x\), we can express this as \(x\) for the first integer and \(x + 1\) for the second integer. These expressions give us a simple and powerful way to work with numbers that follow a specific pattern.
One of the benefits of algebraic expressions is that they allow us to see relationships between different quantities. They are like a toolbox for solving problems, where you can use a variable to stand in place of numbers that can change. This flexibility makes algebraic expressions a key tool in modern mathematics.

• Articulate mathematical ideas clearly.
• Use variables to represent unknown quantities.
• Connect expressions to real-world scenarios.

Simplifying Expressions

Simplifying expressions is an essential part of algebra. It involves reducing an expression to its simplest form where no further arithmetic operations can be performed. In our example, we started with the expression \(20 + (x + 1)\). To simplify, we first removed parentheses: \(20 + x + 1\).
Then, we combined like terms, which are terms that contain the same variable or constants. Here, the like terms were the numbers \(20\) and \(1\), which we combined to get \(21\), leaving us with the final simplified form \(x + 21\).
Understanding how to simplify expressions helps in solving equations more efficiently and makes working with complex expressions more manageable. Simplification means following these steps:

• Remove or distribute parentheses.
• Combine like terms.
• Reorder terms if necessary to facilitate understanding.

Integer Operations

Integer operations are the basic mathematical operations of addition, subtraction, multiplication, and division applied to whole numbers. In the context of our problem, integer operations help us combine numbers and expressions smoothly. For instance, when adding 20 to the second consecutive integer expressed as \(x + 1\), we used addition to merge the numbers and the variable parts into a single expression.
Effective integer operation skills are the foundation upon which simplifying more complex algebraic expressions is built. By mastering integer operations, students can more confidently tackle any mathematical challenge that involves combining different numbers and terms. It's beneficial to practice:

• Understanding how to add and subtract integers consistently.
• Recognizing patterns or sequencing in integers, like consecutive numbers.
• Knowing how integer operations affect expressions and equations.