Problem 63 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ 8(-3)-1 $$

Step-by-Step Solution

Answer

-25

Step 1: Multiplication

Evaluate the expression inside the parentheses first. Multiply 8 by -3. Using the rule of multiplication, \( 8 \times (-3) = -24 \)

Step 2: Subtraction

Subtract 1 from the result of step 1. \(-24 - 1 = -25 \)

Key Concepts

Multiplication Rules

When multiplying numbers, it's essential to keep in mind some simple rules, especially when negative numbers are involved. Here's a quick guide to help you:

• Positive x Positive = Positive: When you multiply two positive numbers, the result is always positive. For example, \(8 \times 3 = 24\).
• Positive x Negative = Negative: Multiplying a positive number by a negative number leads to a negative result. Like in the exercise, \(8 \times (-3) = -24\).
• Negative x Positive = Negative: Similar to the previous rule, changing the position of the negative sign doesn't affect the result. For instance, \((-8) \times 3 = -24\).
• Negative x Negative = Positive: Multiplying two negative numbers results in a positive number. For example, \((-8) \times (-3) = 24\).

This consistency in rules helps keep your calculations straightforward. Knowing these rules makes it easier to handle operations involving both positive and negative numbers.

Order of Operations

To solve any algebraic expression correctly, it's crucial to follow the Order of Operations. A common way to remember the correct order is using the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• M&D: Multiplication and Division (from left to right)
• A&S: Addition and Subtraction (from left to right)

Using PEMDAS ensures that you simplify expressions in the correct sequence. In the given exercise, we followed these rules:

First, we performed the multiplication inside the parentheses:
\(8 \times (-3) = -24\)

Then, we executed the subtraction of 1:
\(-24 - 1 = -25\)

By following these steps, we prevent any mistakes and get the correct result.

Negative Numbers

Handling negative numbers properly is fundamental in algebra. Here are some critical points to remember:

• Negative with Negative: When you subtract a negative number, it's the same as adding its positive counterpart. Hence, \(-(-4) = 4\).
• Adding Negative Numbers: When you add a negative number, you move left from zero on the number line. For instance, \(5 + (-3) = 2\).
• Subtracting Negative Numbers: If you're subtracting a negative number, it becomes more positive. For example, \(7 - (-2) = 9\).

In our exercise, we encountered multiplication and subtraction with negative numbers:

First, we multiplied 8 by -3, resulting in -24, showcasing a positive times negative operation.
\(8 \times (-3) = -24\)

Then, we subtracted 1 from -24, which moved us further left on the number line:
\(-24 - 1 = -25\)

Grasping these concepts will help you navigate algebraic expressions involving negative numbers with ease.