Problem 111 - Elementary and Intermediate Algebra

Question

a. \((-7-4)(-2)\) b. \((-7-4)-2\)

Step-by-Step Solution

Answer

a: 22; b: -13

Step 1: Understand the Expression in Part a

The expression given in part a is \((-7-4)(-2)\). This involves first simplifying the expression inside the parenthesis and then multiplying the result.

Step 2: Simplify the Expression Inside the Parentheses for Part a

Start with the expression inside the parenthesis: \((-7-4)\). Simplifying this gives \((-7-4 = -11)\).

Step 3: Multiply the Simplified Result By -2 for Part a

Now, multiply the simplified result, \(-11\), by \(-2\). The product of two negative numbers is positive, so \(-11 \times -2 = 22\).

Step 4: Understand the Expression in Part b

The expression for part b is \((-7-4)-2\). This involves simplifying the expression inside the parentheses first and then subtracting 2.

Step 5: Simplify the Expression Inside the Parentheses for Part b

Just like in part a, simplify the expression inside the parenthesis: \((-7-4)\). This simplifies to \(-11\).

Step 6: Subtract 2 From the Simplified Result for Part b

Now subtract 2 from \(-11\). This is done by adding a negative: \(-11 - 2 = -11 + (-2) = -13\).

Key Concepts

Multiplication of Integers

When we talk about the multiplication of integers, we're dealing with whole numbers, which include both positive and negative numbers. Here are some key points to keep in mind:

• When you multiply two positive numbers, the result is positive.
• Multiplying a positive number by a negative number gives a negative result.
• Multiplying two negative numbers results in a positive number.

These simple rules stem from the basic properties of numbers and help us simplify expressions accurately. For instance, in exercise part a, we have to multiply (-11) by (-2). Since both numbers are negative, their product will be positive. This is a fundamental aspect often encountered in algebraic expression simplification.

Properties of Negative Numbers

Understanding negative numbers is essential, particularly when they appear in algebraic expressions. Let's break down the basics:

• Negative numbers are less than zero and appear with a minus sign (-).
• When subtracting a negative number, it's like adding a positive. For example, subtracting (-2) is equivalent to adding 2.
• If you have an expression like -(-7), it simplifies to 7, because the two negatives cancel each other out.

These properties guide how operations involving negative numbers are performed and understood. In part b of our exercise, we convert subtraction into adding a negative, which helps us simplify the expression and arrive at the solution -13.

Basic Arithmetic Operations

The building blocks of mathematics are basic arithmetic operations, including addition, subtraction, multiplication, and division.

• When adding numbers, combine their values. For negative numbers, add their absolute values, keeping the sign.
• In subtraction, think of it as adding a negative. For instance, 5 - 2 is the same as 5 + (-2).
• Multiplication finds the total of groups of a number, while division shares equally among groups.

In the given exercise, these operations are essential. Simplifying (-7-4) demonstrates addition of negatives, giving -11. This number is then used in both multiplication and subtraction scenarios, showcasing the versatility of basic arithmetic.