Problem 65 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \((12-14)(1-4)\)

Step-by-Step Solution

Answer

6

Step 1: Simplify inside the parentheses

First, simplify the expressions inside the parentheses. Calculate each one separately.For \(12 - 14\): \(12 - 14 = -2\).For \(1 - 4\): \(1 - 4 = -3\).

Step 2: Multiply the simplified results

Now multiply the results from Step 1.The calculation will be: \(-2 \times -3 = 6\).

Key Concepts

Parentheses in Algebra

Parentheses help us to group parts of an expression together. They tell us which operations to perform first. In algebra, the inside of the parentheses must be simplified before dealing with the rest of the expression. For example, in the expression \( (12-14)(1-4) \), we first solve inside the parentheses. So, we calculate \( 12 - 14 \) and \( 1 - 4 \). This makes our expression easier to handle.

Always remember:

• Simplify inside the parentheses first.
• Perform any operations inside before tackling outside terms.
• If there are multiple levels of parentheses, start with the innermost ones and work outward.

This step-by-step approach prevents mistakes and makes it easier to see what needs to be done next.

Integer Operations

When working with integers, it's important to understand the basic rules of addition and subtraction. Integers are whole numbers, and they can be positive, negative, or zero. In our example, we had \( 12 - 14 \) and \( 1 - 4 \), which involve subtracting two numbers:

• \( 12 - 14 = -2 \)
• \( 1 - 4 = -3 \)

Subtraction with integers can be thought of as adding a negative. For example, instead of \( 12 - 14 \), think \( 12 + (-14) \). This makes it easier to understand that we are moving to the left on the number line.

It's also important to be careful with the signs. If you subtract a bigger number from a smaller number, the result will be negative. This foundational understanding of integer operations is critical for solving algebraic expressions correctly.

Multiplication of Negative Numbers

Multiplying negative numbers can be tricky, but there are simple rules to follow. When two negative numbers are multiplied, the result is positive. This might seem confusing at first, but here's why:

• A negative sign indicates the opposite direction on the number line.
• Multiplying two negatives means turning in the opposite direction twice, which brings us back to the positive side.

In the example \( -2 \times -3 \), we can see this rule in action. Despite both numbers being negative, their product is positive: \( -2 \times -3 = 6 \).

Always remember:

• Negative × Negative = Positive.
• Negative × Positive = Negative.
• Positive × Positive = Positive.

Using these rules will help avoid common pitfalls and make algebraic multiplication much easier to understand and perform.