Problem 89 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the value of each expression. $$|-9-(-3+7)|-|-17-(-2)|$$

Step-by-Step Solution

Answer

-2

Step 1: Simplify Inside the Absolute Value Expressions

First, simplify the expressions inside the absolute values: \(-9-(-3+7)\) simplifies to \(-9 - 4 = -13\) and \(-17-(-2)\) simplifies to \(-17 + 2 = -15\). The expressions then become \(|-13|\) and \(|-15|\).

Step 2: Calculate Absolute Values

Next take the absolute values of -13 and -15, which are 13 and 15, respectively. So the expression becomes \(13 - 15\).

Step 3: Subtract Absolute Values

Finally, subtract 15 from 13, which gives \(-2\).

Key Concepts

Expression Simplification

Expression simplification is the process of altering an expression to make it easier to handle. This involves combining like terms, removing parentheses, and performing arithmetic operations. For example, in the exercise \(-9-(-3+7)\), first simplify the terms inside the parentheses. We perform the operation \(-3 + 7\), which equals 4.

Thus, \(-9 - 4\) simplifies to \(-13\). Simplification helps by making complex expressions more manageable, allowing you to see underlying patterns and relationships more clearly.

• Combine like terms.
• Remove parentheses where necessary.
• Simplify arithmetic operations to reach a simpler form.

Remember, the goal is to make the expression as straightforward as possible.

Subtracting Integers

Subtracting integers involves thinking about the direction and magnitude of numbers on the number line. When subtracting integers, you actually add the opposite number. For example, \(-17 - (-2)\) becomes \(-17 + 2\).

This process simplifies to \(-15\), since you are effectively moving two steps to the right from \(-17\) on the number line. Subtraction of integers can be tricky, but a simple way to remember it is:

• Subtracting a positive is like adding a negative.
• Subtracting a negative is like adding a positive.

Getting comfortable with these rules helps in evaluating many kinds of expressions efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are core in mathematics for modeling real-world problems and abstract mathematical concepts.

In our exercise, we are working with numbers only, but the concept is the same. You're looking to simplify expressions such as \(-9-(-3+7)\) by breaking them down following the order of operations. This is known as PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

• Understand the role of variables if they are present.
• Apply operations in the correct order.
• Simplify systematically using mathematical rules.

Understanding algebraic expressions is crucial because it develops problem-solving skills and is applicable in various fields.