Problem 10 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$5-(-17)$$

Step-by-Step Solution

Answer

The answer is 22.

Step 1: Write down the given numerical values

The initial problem is \(5 - (-17)\).

Step 2: Simplify the Expression

Since subtracting a negative number is equivalent to adding a positive one, the equation can be rewritten as \(5 + 17\). This simplification is based on the rule of signs in mathematics.

Step 3: Solve the Equation

By adding the two numbers, the solution to the problem is calculated as \(22\).

Key Concepts

Simplifying Expressions

Understanding how to simplify expressions is critical when working with mathematical equations. Simplification often involves combining like terms or applying mathematical operations in a way that makes the expression easier to work with or understand. In the context of the exercise, where we have 5 - (-17), simplification involves changing the operation from subtraction to addition. This is because the subtraction of a negative number is the same as adding its positive counterpart.

When simplifying expressions that involve negative numbers, it's important to remember that two negatives make a positive, so -(-17) becomes +17. This change transforms the expression to 5 + 17, which is straightforward and easier to compute. Keeping simplification principles in mind, such as combining like terms and recognizing operations between positives and negatives, can help students efficiently navigate arithmetic and algebraic problems.

Rule of Signs in Mathematics

The rule of signs is a fundamental principle in mathematics that dictates how the signs of numbers change under various operations. This rule is crucial when simplifying expressions, especially those involving subtraction and addition of negative numbers. To subtract a negative number, like in our exercise 5 - (-17), we must apply the rule of signs. According to this rule, subtracting a negative is the same as adding a positive. Hence, the expression simplifies to 5 + 17.

Key Points of the Rule of Signs:

• Adding positive to positive retains a positive sign. For example, +5 + (+3) = +8.
• Adding negative to negative retains a negative sign. For instance, -5 + (-3) = -8.
• Subtracting negative from positive is like adding positive to positive. As in our exercise, 5 - (-17) = 5 + 17.
• Subtracting positive from negative is like adding another negative, for example, -5 - (+3) = -5 + (-3) = -8.

Understanding this principle is vital for correctly performing operations and simplifying various mathematical expressions.

Adding Positive Numbers

Adding positive numbers is one of the most basic and essential arithmetic skills. This process is intuitive and forms the foundation for more complex mathematical concepts. When we add positive numbers, we are essentially counting up or increasing value. For example, in our original exercise, simplifying 5 - (-17) leads to adding the positive numbers 5 and 17 together.

When adding two positive numbers, the sum will also be positive, which is an inherent trait of addition. The process of adding positive numbers involves combining the individual values and arriving at a total that represents their collective magnitude. This concept is not only crucial for solving basic arithmetic problems, such as the one provided, but it also forms the basis for more advanced calculations, including those found in algebra, calculus, and beyond. Practicing addition with positive numbers helps reinforce a student's understanding of numerical relationships and inherently builds their confidence to tackle more challenging math concepts.