Problem 111 - Introductory Algebra for College Students

Question

Write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The product of \(-9\) and \(-3,\) decreased by \(-2\)

Step-by-Step Solution

Answer

The numerical expression is 29.

Step 1: Interpreting the phrase

First, identify the operation words in the phrase. 'Product' implies multiplication and 'decreased by' implies subtraction. The phrase translates to '(-9) multiplied by (-3) then subtract -2'.

Step 2: Performing the multiplication

Multiply -9 and -3. The multiplication of two negative numbers results in a positive number. Therefore, (-9)*(-3) equals 27.

Step 3: Decreasing the result by -2

Subtract -2 from the result of multiplication. The operation is 27 - (-2). We know that subtracting a negative number is the same as adding the positive value, hence it becomes 27 + 2.

Step 4: Performing the addition

After converting subtraction of -2 into addition operation, perform the addition: 27 + 2, which equals 29.

Key Concepts

Simplifying Numerical Expressions

Understanding how to simplify numerical expressions is essential in algebra. It involves reducing an expression to its simplest form while performing all the operations in the correct order. Think of it as a recipe: just as you would add ingredients step-by-step following a recipe, you simplify expressions by following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

To simplify the numerical expression from our exercise, you'd multiply first (since multiplication comes before subtraction) and then subtract. After these operations, you reach a single number, the simplest form of the original expression.

Operations in Algebra

Algebra is filled with various operations that tell us what to do with numbers. The four basic algebraic operations are addition, subtraction, multiplication, and division. In our example, the operations involve multiplication and subtraction.

When working with algebraic expressions, it's crucial to understand these operations and how they interact. For example, knowing that multiplication of negative numbers leads to a positive result can spare you from common mistakes. Additionally, when combining these operations, pay close attention to the signs to ensure accurate calculations.

Multiplication of Negative Numbers

The rule for the multiplication of negative numbers may seem counterintuitive at first, but it's essential for solving algebraic equations correctly. The rule is simple: when you multiply two negative numbers, the result is always positive. This is because you are essentially reversing a reversal, landing you back in positive territory.

In the context of our textbook example, when you multiply \( -9 \) by \( -3 \) you get \( 27 \) because both numbers have the 'negative' sign, which cancel each other out. Remembering this rule is key to simplifying expressions correctly.

Subtraction of Negative Numbers

Subtracting negative numbers often trips up students, but it’s like discovering a double negative in a sentence—it turns positive! When you subtract a negative number, you're essentially adding its positive counterpart.

If we revisit our example where we have \( 27 - (-2) \), subtracting \( -2 \) is the same as adding \( 2 \). This adjustment changes the operation and leads you to the correct answer, \( 29 \). This principle is a fundamental aspect of algebra and simplifying numerical expressions.