Problem 86 - Intermediate Algebra

Question

In the following exercises, simplify each expression. (a) \(35-16\) (b) \(35+(-16)\)

Step-by-Step Solution

Answer

Both (a) and (b) simplify to 19.

Step 1: Simplify the expression for part (a)

The expression given is 35 - 16. To simplify it, perform the subtraction: \(35 - 16 = 19\). Therefore, the simplified form of the expression for part (a) is 19.

Step 2: Simplify the expression for part (b)

The expression given is 35 + (-16). To simplify it, think of adding a negative number as subtracting the positive version of that number. So, \(35 + (-16) = 35 - 16\). As we calculated earlier, \(35 - 16 = 19\). Therefore, the simplified form of the expression for part (b) is 19.

Key Concepts

Integer Operations

Integer operations are basic arithmetic tasks involving whole numbers, which can be either positive or negative. Understanding these operations is essential for simplifying mathematical expressions effectively. Here are the main points to remember about integer operations:

• Positive numbers are greater than zero.
• Negative numbers are less than zero.
• Zero is neutral.

Integer operations include addition, subtraction, multiplication, and division. When dealing with these operations, it's crucial to follow the rules for combining positive and negative values. For instance, when you add a positive integer to a negative integer, you are effectively subtracting the smaller number from the larger number.

Addition and Subtraction

When simplifying expressions that involve addition and subtraction, it's important to understand how these operations interact with integers. Here’s a detailed breakdown:

• Addition: Adding two positive numbers always gives a positive result, and adding two negative numbers always gives a negative result.
• Subtraction: Think of subtraction as adding the opposite. For instance, subtracting a positive number is equivalent to adding its negative counterpart. This helps simplify expressions involving negative numbers.

Using the example from our exercise:

For the expression \(35 - 16\), we subtract 16 from 35, which gives us 19.

For the expression \(35 + (-16)\), we add 35 and -16. Since adding a negative is like subtracting a positive, this is also equivalent to \(35 - 16\), resulting in 19.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign (-). They can often be a source of confusion, especially when combined with positive numbers or other negative numbers in arithmetic operations. Here are the essential concepts to keep in mind:

• Negative numbers are used to represent values below zero, such as temperatures below freezing or depths below sea level.
• When subtracting a negative number, it becomes an addition (e.g., \(5 - (-3) = 5 + 3\)), because subtracting a negative is equivalent to adding a positive.
• When adding a negative number, it is like subtracting the absolute value of the number (e.g., \(7 + (-2) = 7 - 2\)).

Using these rules, we can see why \(35 + (-16)\) is the same as \(35 - 16\). Understanding how to manipulate negative numbers ensures you can simplify expressions correctly and confidently.