Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

Use an associative property to rewrite each algebraic expression. Once the grouping has been changed, simplify the resulting algebraic expression. $$9+(3+x)$$

Step-by-Step Solution

Answer

Using the associative property, the expression \(9 + (3 + x)\) simplifies to \(12 + x\).

Step 1: Apply Associative Property

Firstly, apply the associative property to the given algebraic expression. Hence, the expression \(9 + (3 + x)\) gets converted into \((9 + 3) + x\). This is done to rearrange the numbers and simplify calculations.

Step 2: Simplify the Expression

Now, sum up the numbers that are grouped together. So, \((9 + 3) + x\) will become \(12 + x\).

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase combining numbers, variables, and operators. It does not have an equal sign, which distinguishes it from equations. For example, in our exercise, the expression is given as \(9 + (3 + x)\). Here, you see numbers (9 and 3), a variable \(x\), and an operator, which is addition. The beauty of algebraic expressions is their flexibility. Their structure can be modified using different properties like associative and distributive properties to make calculations easier. Understanding algebraic expressions is essential because they form the basis for solving more complex mathematical problems. Remember, expressions don't have solutions unless they're set equal to something.

Simplifying Expressions

Simplifying expressions involves writing them in their most compact or efficient form without changing their value. The goal is to make the expression easier to read or work with. In our exercise, the expression \(9 + (3 + x)\) was simplified to \(12 + x\).

To simplify an expression, you often need to use properties of arithmetic, like the associative property. This property allows you to rearrange brackets as needed, as long as the order of the numbers remains the same. Simplification might also involve combining like terms - terms that have the same variable raised to the same power. Simplifying doesn't solve the expression but makes it more understandable and prepares it for further operations if needed.

Order of Operations

Order of operations is a set of rules that tells us the correct sequence in which to solve different parts of a mathematical expression. To ensure consistency, this order is crucial. A common way to remember the order is PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In the given expression \(9 + (3 + x)\), parentheses indicate which operations to perform first. By applying the associative property, we rearrange the parentheses to simplify differently. This doesn't change the order of operations but shifts how we simplify. Following PEMDAS ensures that no matter how complex an expression may appear, any simplification or operation will lead to the same result if performed correctly.