Problem 77 - Algebra A Combined Function

Question

Write each algebraic expression described. Simplify if possible. See Example \(11 .\) If \(x\) represents the first of two consecutive odd integers, express the sum of the two integers in terms of \(x\).

Step-by-Step Solution

Answer

The sum of the two consecutive odd integers is \(2x + 2\).

Step 1: Define the Integers

We need to find expressions for two consecutive odd integers. Let's say the first integer is defined by \(x\). Since the integers are consecutive odd numbers, the next integer will be \(x + 2\).

Step 2: Express the Sum of Integers

To find the sum of these two consecutive odd integers, we add them together. The expression becomes: \(x + (x + 2)\).

Step 3: Simplify the Expression

Simplify the expression obtained in the previous step: \(x + x + 2 = 2x + 2\).

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order without any gaps. In a mathematical sense, these numbers are sequential, each number increasing by one from the previous number. For example, 3, 4, 5 are consecutive integers. In the context of algebra, we often define the first integer as a variable, usually denoted by \(x\). Subsequently, the next consecutive integer could be \(x+1\). This concept becomes handy when solving problems involving a sequence of integers. It allows us to structure equations and expressions that capture the quality of being consecutive in a logical manner.

Odd Integers

Odd integers are simply integers that cannot be divided evenly by 2. They end with one of the digits 1, 3, 5, 7, or 9 in the decimal system. These numbers are integral and, when paired together in sequence, they form consecutive odd integers. For example, 3 and 5 are consecutive odd integers. In algebra, when defining consecutive odd integers, if the first odd integer is \(x\), the next consecutive odd integer will be \(x+2\). This is because the difference between any two sequential odd numbers is always 2. Identifying and properly defining odd integers in algebraic expressions is a crucial step in solving many mathematical problems.

Summing Expressions

Summing expressions in algebra involves adding different quantities together. This operation is foundational in algebra as it combines terms into a single expression.

• For example, if you have two expressions, \(a\) and \(b\), their sum is \(a + b\).
• Another example includes summing identical terms, such as \(3x + 2x\), which equals \(5x\).
• When dealing with consecutive odd integers, if the first integer is \(x\) and the second is \(x+2\), their sum is expressed as \(x + (x+2)\).

Recognizing how to effectively construct and comprehend summing expressions is a crucial algebraic skill. It ensures that you can properly apply arithmetic operations across expressions.

Expression Simplification

Simplification is the process of reducing an algebraic expression to its simplest form. It means combining all like terms and making the expression as concise as possible.

• Like terms, such as \(x\) terms and constant terms, are combined together.
• In our example, the expression \(x + x + 2\) was simplified to \(2x + 2\).
• This happens because \(x + x\) equals \(2x\), and \(+ 2\) is a constant that remains as it is.

Expression simplification is a key skill when dealing with equations. It helps to provide a clear and precise representation of an algebraic expression, which in turn makes further mathematical manipulations easier.