Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\((8-17)+3\)

Step-by-Step Solution

Answer

\(-6\)

Step 1: Identify Parentheses

The first operation to perform is the one inside the parentheses, which is subtracting 17 from 8.

Step 2: Perform subtraction

Subtract 17 from 8, which gives \(-9\)

Step 3: Add 3

Add 3 to \(-9\), which is equal to \(-6\)

Key Concepts

Order of Operations

When simplifying algebraic expressions, it's crucial to follow the Order of Operations. This set of rules dictates the sequence in which operations should be carried out to ensure accurate results. The acronym PEMDAS is often used to remember the order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Let's break it down further:

• **Parentheses**: Always perform operations inside parentheses first. This ensures that calculations within grouped terms are completed before moving on to other operations.
• **Exponents**: Solve any exponents next, though they don't appear in this particular exercise.
• **Multiplication and Division**: These are done next, from left to right. However, our problem doesn't include these operations.
• **Addition and Subtraction**: Finally, perform addition and subtraction as they appear from left to right.

In the exercise given, by subtracting first due to parentheses, and then adding, we correctly follow the Order of Operations.

Subtraction in Algebra

Subtraction in algebra involves more than merely taking one number away from another. It sets the stage for understanding negative numbers and their properties.

Consider the exercise, \((8-17)+3\):

• First, identify that you're subtracting a larger number (17) from a smaller one (8), which will yield a negative result.
• When calculating \(8-17\), think of it as moving 17 units left from 8 on a number line, resulting in \(-9\).

This step highlights the negative integer outcome of subtraction and prepares the ground for further operations. Remember that subtraction can be viewed as the addition of a negative.

Addition of Integers

Adding integers, especially when they include negative numbers, is a vital skill.
When adding a positive integer to a negative one, think of it as making a movement on the number line.

In the problem, we take the result of \(8-17\) which is \(-9\), and add 3:

• Visualize yourself starting at \(-9\) on the number line.
• Then move 3 units to the right, ending at \(-6\).

Whenever you're combining a positive and negative number, the number with the larger magnitude determines the sign of the result. So, \(-9 + 3\) results in \(-6\), as \( -9 \) has a larger absolute value.