Problem 106 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ (6-11)(8-13) $$

Step-by-Step Solution

Answer

-5 \times -5 = 25

Step 1: Simplify Inside the Parentheses

Calculate the value inside each set of parentheses. For \(6 - 11\), subtract 11 from 6 to get \(-5\). For \(8 - 13\), subtract 13 from 8 to get \(-5\). So the expression becomes \((-5)(-5)\).

Step 2: Multiply the Simplified Values

Multiply the two simplified values: \(-5 \times -5\). Since the product of two negative numbers is positive, \(-5 \times -5 = 25\).

Key Concepts

Multiplying Integers

Multiplying integers is an essential skill in algebra. Here are some quick points to remember:

• When you multiply two positive integers, the result is always positive.
• When you multiply a positive integer by a negative integer, the result is always negative.
• When you multiply two negative integers, the result is always positive.

For example, \( -5 \times -5 \) equals 25 because the product of two negative numbers is positive. This is a useful rule to keep in mind when simplifying algebraic expressions.

Parentheses in Algebra

Parentheses are an important part of algebra. They indicate which operations should be performed first. Think of them as a way to group parts of an expression that need to be simplified together.

For example, in the expression \( (6-11)(8-13) \), you are required to first simplify the expressions within the parentheses.

This means calculating \( 6 - 11 \) and \( 8 - 13 \) before doing anything else. Once simplified, you have \( -5 \times -5 \) which then equals 25. Always resolve the values inside the parentheses first to avoid mistakes.

Negative Numbers

Understanding negative numbers is crucial in algebra. Here is a little guide to help you:

• Negative numbers are numbers less than zero.
• When adding a negative number, it is equivalent to subtraction.
• When subtracting a negative number, it is equivalent to addition.

For instance, subtracting 13 from 8 gives you a negative number: \( 8 - 13 = -5 \). Similarly, subtracting 11 from 6 results in \( 6 - 11 = -5 \). Keep these rules in mind when working with negative numbers in algebraic expressions.