Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each series of additions and subtractions. $$-0.16-5.2-(-0.87)$$

Step-by-Step Solution

Answer

The simplified expression is \(-4.49\)

Step 1: Identify the operations

The given expression includes subtraction and addition operations. It's critical to note that subtraction of a negative number is the same as addition. The expression can thus be rewritten as: \(-0.16 - 5.2 + 0.87\).

Step 2: Perform the operations

Next, add and subtract the numbers from left to right, giving: \(-0.16 - 5.2 + 0.87 = -5.36 + 0.87\), and then: \(-5.36 + 0.87 = -4.49\)

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying algebraic expressions. It is a collection of rules that determine which procedure to perform first in a mathematical expression. The standard order, often remembered by the acronym PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the exercise \( -0.16 - 5.2 - (-0.87) \) there are no parentheses or exponents to consider, making it straightforward to proceed with the arithmetic operations. However, the presence of a double negative (subtraction of a negative number) should be recognized as an operation to resolve early, almost like an unwritten step before the left-to-right addition and subtraction.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These are the building blocks of most mathematical calculations. In simplifying algebraic expressions, it's important to perform these operations accurately and in the correct order.

For the given expression, we are dealing with addition and subtraction. A common mistake is ignoring the signs of the numbers involved. To avoid this, we rewrite subtractions as additions where possible (Step 1 in the solution), and we then combine the terms carefully, keeping track of the negative and positive values (Step 2), leading to the final answer \( -4.49 \).

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). Understanding how to work with negative numbers is fundamental when simplifying expressions, especially those involving subtraction as seen in the exercise.

Subtracting a negative number is one of the most common operations that confuse students. It helps to remember that two negatives make a positive, so when you see a minus sign followed by a negative number, you are actually adding the positive value of that number. In this exercise, the subtraction of \( -0.87 \) is essentially the addition of \( 0.87 \), which is addressed in the solution by transforming it into an addition operation.