Problem 124 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate both expressions for \(x=4 .\) What do you observe? $$3(x+5) ; 3 x+15$$

Step-by-Step Solution

Answer

Both expressions \(3(x+5)\) and \(3x+15\) evaluate to 27 when x=4. This shows that the expressions are equivalent.

Step 1: Evaluate the First Expression

Substitute the value of x, which is 4, in the first expression that is \(3(x+5)\). This changes the expression to \(3(4+5)\). After performing the operation inside the brackets first (4+5), the expression becomes \(3(9)\) and then the multiplication gives 27.

Step 2: Evaluate the Second Expression

Now substitute the value of x in the second expression that is \(3x+15\). This changes the equation to \(3*4+15\). Perform the multiplication first to get \(12+15\) due to the order of operations in mathematics. The result of this operation is 27.

Step 3: Analyze the Results

Based on the evaluation of both expressions, it is observed that the results from both expressions are the same.

Key Concepts

Evaluating Expressions

Evaluating expressions involves replacing variables in an equation with specific numbers and then simplifying to find the result. Let's consider the first expression given in the exercise: \(3(x+5)\). Here, we substitute \(x\) with the value 4. This transforms the expression into \(3(4+5)\). By solving the expression inside the parentheses, we get \(3 \times 9\), which equals 27.

The process is similar for the second expression, \(3x+15\). Substitute \(x = 4\) to obtain \(3 \times 4 + 15\). Calculate \(3 \times 4\) to get 12 and then add 15 for a final result of 27.

It's important to note that evaluating expressions involves a straightforward substitution and follows arithmetic operations in sequence according to a set mathematical order.

Simplification

Simplification is the process of reducing an expression to its simplest form. In the context of the exercise, after substituting \(x = 4\) into both expressions, the focus is on simplifying them. For \(3(x+5)\), after substitution, it's simplified by first dealing with the term inside the parentheses: \(4+5\). This results in 9, which, when multiplied by 3, gives 27.

For \(3x+15\), simplification begins after substitution by performing the calculations \(3 \times 4\), resulting in 12, and then adding 15. The expression streamlines to a single value, 27.

• Always perform operations within parentheses first.
• Combine like terms, and reduce expressions step-by-step.

Simplification helps in making complex expressions easier to handle and understand by confirming their values match when different forms of the same expression are evaluated.

Order of Operations

Order of operations is a fundamental mathematical rule that dictates the sequence in which calculations should be performed to ensure consistent results. This rule is often remembered using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In the exercise's first expression \(3(x+5)\), the rule guides us to first resolve the parentheses \(4+5\), resulting in 9, then perform multiplication by 3.

For the second expression \(3x+15\), substitute first, then follow with multiplication \(3 \times 4\), which yields 12 before finally performing the addition to get 27.

• Start with the operations inside parentheses.
• Proceed with exponents, followed by multiplication and division.
• Finish with addition and subtraction.

The order of operations is crucial not only for maintaining standardization in mathematical expression evaluations but also for avoiding mistakes and ensuring the correct outcome.