Problem 67 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \((7-10)(10-4)\)

Step-by-Step Solution

Answer

-18

Step 1: Simplify the expression inside the first parenthesis

Calculate the result of the expression inside the first parenthesis \(7 - 10\). This simplifies to \( -3\).

Step 2: Simplify the expression inside the second parenthesis

Calculate the result of the expression inside the second parenthesis \(10 - 4\). This simplifies to \( 6\).

Step 3: Multiply the simplified values from Step 1 and Step 2

Multiply the results from Step 1 and Step 2. So, \((-3)\times(6) = -18\).

Key Concepts

Simplifying Expressions Inside Parentheses

When simplifying algebraic expressions, it's crucial to first focus on the innermost parentheses. Parentheses indicate that any operation inside them should be performed before anything else in the equation. For the given problem, the expression inside the first parenthesis is \(7 - 10\). We need to subtract 10 from 7.

Since 10 is larger than 7, we get \(7 - 10 = -3\). This is a fundamental step, as simplifying expressions inside parentheses helps streamline the rest of the problem.

Next, let's look at the second parenthesis: \(10 - 4\). Here, we subtract 4 from 10, which gives us \(10 - 4 = 6\). Again, simplifying within parentheses first makes the entire process more manageable.

Understanding Integers

Integers are whole numbers that can be positive, negative, or zero. In the context of our problem, integers help us navigate operations with both negative and positive values.

For example, when we calculate \(7 - 10\), we are essentially dealing with two integers: 7 and -10. We adjust accordingly by understanding that subtracting a larger number from a smaller one results in a negative integer.

Similarly, positive integers like 6 (as found in \(10 - 4\)) remain straightforward. Knowing how to handle integers is essential for both simplifying intermediate steps and accurately solving the final operation.

Multiplication of Negative and Positive Numbers

Multiplying numbers can sometimes be tricky, especially when dealing with both negative and positive values. The main rule to remember is:

• The product of two positive numbers or two negative numbers is positive.
• The product of a positive number and a negative number is negative.

In the final step of our problem, we need to multiply -3 (from \(7 - 10\)) and 6 (from \(10 - 4\)).

Using the rules above:

\((-3) \times (6)\)

This gives us -18, since a negative number multiplied by a positive one results in a negative outcome.

Understanding these rules makes it easier to perform multiplication, ensuring accuracy and reducing mistakes in algebra.