Problem 34 - Beginning and Intermediate Algebra

Question

Perform the operation. See Example 3. Subtract 3 from \(-2\)

Step-by-Step Solution

Answer

The result is -5.

Step 1: Understand the Problem

We are asked to subtract 3 from -2. This operation can be set up as \(-2 - 3\).

Step 2: Rewrite as Addition

We know that subtracting a number is the same as adding its opposite. So, \(-2 - 3\) becomes \(-2 + (-3)\).

Step 3: Perform the Addition

Now, perform the addition: \(-2 + (-3) = -5\). This is because adding a negative number increases the negativity of the original number.

Key Concepts

Subtraction of Integers

Subtracting integers may seem complex at first, but it becomes much easier once you understand the basic rules. When we subtract one integer from another, we are essentially determining the distance between the two numbers on a number line.

A useful strategy is to think of subtraction in terms of addition. Instead of subtracting a number, you can add its opposite. For example, subtracting 3 from \(-2\) is the same as adding the opposite of 3, which is \(-3\). This turns the original expression \(-2 - 3\) into \(-2 + (-3)\).

By converting subtraction operations into addition, working with integers becomes more intuitive. This leads us directly into our next topic: addition of integers.

Addition of Integers

Adding integers involves combining values to see how they change the position on a number line. When integers have the same sign, you simply add their absolute values.

For example, when adding \(-2\) and \(-3\), the absolute values are 2 and 3. Simply add:

• Step 1: Take the absolute value of both integers, which makes 2 and 3.
• Step 2: Add these values together: 2 + 3 = 5.
• Step 3: Since both integers are negative, keep the sign negative: \(-5\).

When dealing with integers of different signs, like \(-2\) + 3, you subtract the smaller absolute value from the larger one. The sign of the result will match the sign of the integer with the larger absolute value. Understanding these rules becomes important when tackling more complex problems.

Negative Numbers

Negative numbers can sometimes be intimidating, but they follow specific patterns that make them easier to work with once you understand them.

Negative numbers are numbers less than zero, and they are found to the left of zero on the number line. They represent values like debts, below-zero temperatures, or floors below ground level. Here are some key points:

• As you move further left on the number line, numbers become more negative.
• Adding negative numbers increases the 'negative' distance from zero, which makes them appear smaller.
• Subtracting a negative number is equivalent to adding its positive counterpart.

Grasping these basic ideas about negative numbers is essential for solving integer operations accurately.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Unlike an equation, an algebraic expression does not contain an equality sign.

For instance, consider the expression \(-2x + 3\). It combines variables, constants, and operations to represent a value. Here are key components:

• **Constants** are known values, like 3 in the expression.
• **Variables** are symbols like x that represent unknown or changeable values.
• **Coefficients** are numbers multiplied by the variables, such as \(-2\) in \(-2x\).
• **Operations** such as addition and subtraction, illustrate the relationships between terms.

Understanding these elements of algebraic expressions allows you to manipulate and simplify them in complex calculations, making integer operations much simpler to manage.