Problem 79 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(2x + 2\).

Step 1: Identify the consecutive integers

If we are given that \(x\) is the first of four consecutive integers, the integers can be expressed as \(x, x+1, x+2, x+3\). These expressions represent the four consecutive integers starting from \(x\).

Step 2: Identify the required integers

The problem asks for the sum of the first and the third integers from this sequence. The first integer is \(x\) and the third integer is \(x+2\).

Step 3: Formulate the expression for required sum

We now need to write an expression for the sum of the first and third integers. This becomes the arithmetic expression: \(x + (x + 2)\).

Step 4: Simplify the expression

Simplify the expression obtained in Step 3: Combine like terms. \(x + x + 2 = 2x + 2\). This is the simplified expression for the sum of the first and third consecutive integers.

Key Concepts

Consecutive integers

Consecutive integers are simply integers that follow each other in order without any gaps or skips. For example, when you start with a number like 1, the next consecutive integers would be 2, 3, and 4. In algebra, consecutive integers often appear in problems where we need to define a sequence. If we start with an integer, say \(x\), then the next consecutive integers would be \(x+1\), \(x+2\), and so on. This pattern of adding 1 to the previous number helps in solving problems related to integer sequences by creating variables out of everyday numbers.
In the given exercise, we used these to define four consecutive integers starting from \(x\). This approach is powerful because it lets us express patterns and relationships using algebraic expressions.

• The starting point was \(x\).
• Consecutive integers were expressed as \(x, x+1, x+2, x+3\).

This ability to craft sequences using algebra underlies many problems in mathematics.

Simplifying expressions

Simplifying expressions is a key skill in algebra that involves rewriting an expression in its most reduced or simplified form. This process often involves combining like terms, which are terms that have the same variable raised to the same power. For example, in the expression \(x + x + 2\), we observe that there are two \(x\) terms that can be combined.
To simplify, we add the coefficients (the numbers in front of the variables) of like terms:

• Combine the \(x\) terms: \(x + x = 2x\)
• Keep the constant term: \(+2\)

After simplification, the original expression \(x + (x + 2)\) becomes \(2x + 2\). This is a more straightforward form, which is easier to understand and work with. Learning how to simplify expressions smoothly is crucial as it prepares students for more advanced algebra and other areas of mathematics.

Variables in algebra

Variables are symbols used in algebra to represent numbers or values that can change. They are usually denoted by letters such as \(x\), \(y\), or \(z\). The use of variables allows us to formulate general rules and equations that can be applied to a vast range of situations. In our exercise, \(x\) represents the first integer in the sequence of four consecutive integers. Using \(x\) as a starting point, we then derived expressions for subsequent integers.
Variables make it possible to:

• Represent unknown values, which we may later solve for.
• Create expressions that model real-world situations.
• Explore relationships between different quantities.

Mastering the use of variables is fundamental in algebra. They act as the backbone of algebraic operations and expressions, providing the flexibility to generalize solutions and tackle diverse mathematical problems.