Problem 150 - Introductory Algebra for College Students

Question

Exercises 150–152 will help you prepare for the material covered in the next section. Evaluate \((3 x-1)(x+2)\) for \(x=\frac{1}{3}\)

Step-by-Step Solution

Answer

The solution is 0.

Step 1: Substitute the given value

Substitute \(\frac{1}{3}\) for \(x\) in the expression \((3x-1)(x+2)\), to obtain \((3*\frac{1}{3}-1)(\frac{1}{3}+2)\).

Step 2: Simplify

Simplify inside each set of parentheses, which results in \((1-1)(\frac{1}{3}+2)\) or \((0)(2.333)\).

Step 3: Apply multiplying operation

Now, multiply these results together. Any number multiplied by zero equals zero, so the answer is 0.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). The purpose of algebraic expressions is to denote relationships and to allow a systematic way of solving problems that deal with numbers and quantities. Algebra is known for its use of symbols, which enable mathematicians and students alike to easily manipulate and solve for unknowns.

For example, the expression \(3x - 1)(x + 2)\) introduced in the exercise above involves two variables multiplied by each other. It is a more advanced form since it includes not only algebraic addition and subtraction but also the multiplication of two binomials.

Substitution Method

The substitution method is a fundamental algebraic technique used to evaluate expressions. It involves replacing a variable in an expression with a specific value. This is a powerful tool when solving equations, as it allows you to isolate and solve for one variable at a time. When you substitute a value into an algebraic expression, you effectively make it a numerical expression, which can then be simplified to find a solution.

In the exercise, we're given the algebraic expression \(3x - 1)(x + 2)\) and asked to evaluate it for \(x = \frac{1}{3}\). Substitution is done by replacing every instance of \(x\) in the expression with \(\frac{1}{3}\). Always ensure the substituted value is clearly placed within the expression to avoid any calculation errors. After substitution, you then proceed to simplify the numerical expression to find the final value.

Simplifying Expressions

Simplifying an expression in algebra means to reduce it to its simplest form. This usually involves performing all possible operations, combining like terms, and reducing any fractions. The objective is to make the expression as straightforward as possible while maintaining its original value.

Looking at the solution provided for our exercise, after the substitution step, you saw an expression with numbers ready to be simplified. The expression \((1 - 1)(\frac{1}{3} + 2)\) simplifies first within each parenthesis. The first set simplifies to 0 because 1 - 1 is 0. The second set of parentheses simplifies to \((\frac{1}{3} + \frac{6}{3})\), or 2.333 when added together.

Once you simplified each part, you multiply the results as instructed, obtaining a final answer of 0. It's important when simplifying to take one step at a time, ensuring each operation is performed accurately. This systematic approach helps to prevent mistakes and ensures an accurate conclusion to the problem at hand.