Problem 82 - Algebra A Combined Function

Question

Write each algebraic expression described. Simplify if possible. See Example \(11 .\) Two sides of a quadrilateral have the same length, \(x\) while the other two sides have the same length, both being the next consecutive odd integer. Write the sum of these lengths. Then simplify this expression.

Step-by-Step Solution

Answer

The sum of the side lengths is \(4x + 4\).

Step 1: Identify the Known Sides

We are given that two sides of the quadrilateral have the same length, denoted as \(x\). So the lengths of these two sides are \(x\).

Step 2: Identify the Unknown Sides

The problem states that the next two sides are the next consecutive odd integers. If one side is \(x\), an odd integer can be represented as \(x\) when it's odd, and its consecutive odd integer is \(x + 2\), if \(x\) itself is odd. Thus, both sides have length \(x + 2\).

Step 3: Write the Expression for the Perimeter

The expression for the perimeter of the quadrilateral is the sum of all its sides: two sides of length \(x\), and two sides of length \(x + 2\). This gives the expression: \(x + x + (x + 2) + (x + 2)\).

Step 4: Simplify the Expression

Combine like terms in the expression \(x + x + (x + 2) + (x + 2)\): \[ x + x + x + 2 + x + 2 = 4x + 4 \]This simplified expression represents the sum of the lengths of the sides of the quadrilateral.

Key Concepts

Quadrilateral Perimeter

Understanding the perimeter of a quadrilateral involves summing the lengths of its four sides. When two pairs of sides are equal, this concept becomes simpler, as in this exercise. Each quadrilateral, by definition, is a four-sided polygon. To find its perimeter, you just add up the lengths of all these sides.
In our scenario, two sides have the same length, represented by the variable \(x\). To complete the quadrilateral, the other two sides are consecutive odd integers starting from \(x\). Remembering that odd integers differ by 2 is crucial here, making the consecutive odd integer \(x + 2\).
Thus, for the quadrilateral:

• Two sides are \(x\).
• The other two sides are \(x + 2\).

Add them together to find the perimeter: \(x + x + (x + 2) + (x + 2)\), resulting in the initial expression for perimeter.

Simplifying Expressions

Simplifying algebraic expressions is a vital skill in algebra that involves combining like terms to make the expression easier to work with. When simplifying, look for terms that involve the same variable with the same power.
In our initial expression for the quadrilateral's perimeter, \(x + x + (x + 2) + (x + 2)\), you can see that you have multiple \(x\) terms. The key here is combining these like terms to simplify the expression.
Here's how to simplify step by step:

• Combine the \(x\) terms: \(x + x + x + x\), which adds up to \(4x\).
• Combine the constants: \(2 + 2\), which adds up to 4.

This gives you the simplified expression: \(4x + 4\). Breaking it down this way makes the expression much more manageable for further calculations.

Consecutive Integers

Consecutive integers are numbers that follow each other in sequence, differing by one for regular consecutive numbers and by two when dealing with odd or even sequences. Here, we're focusing on consecutive odd integers.
Let's say we start with an odd integer \(x\). The next consecutive odd integer would be \(x + 2\). This pattern holds no matter what the starting point is, as long as \(x\) is odd. This systematic approach makes it easy to derive expressions involving these integers.
In our quadrilateral exercise:

• One pair of sides is \(x\), an odd integer.
• The other pair is \(x + 2\), the consecutive odd integer.

Recognizing and working with consecutive integers is a crucial algebra concept. It allows you to quickly build expressions and solve real-world problems involving sequences.