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Chapter 1: Problem 89

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

Short Answer

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Step by step solution

01

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 \(|-15|\).
02

Calculate Absolute Values

Next take the absolute values of -13 and -15, which are 13 and 15, respectively. So the expression becomes \(13 - 15\).
03

Subtract Absolute Values

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, use the given formula to perform the indicated operation with the two fractions. $$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b} ; \frac{9}{17}-\frac{5}{17}$$

Describe what it means to raise a number to a power. In your description, include a discussion of the difference between \(-5^{2}\) and \((-5)^{2}\)

Explain how to simplify an algebraic expression in which a negative sign precedes parentheses.

I read that a certain star is \(10^{4}\) light-years from Earth, which means \(100,000\) light-years. When I evaluated \((-1)^{n},\) I obtained positive numbers when \(n\) was even and negative numbers when \(n\) was odd

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The difference between \(-6\) and the quotient of 12 and \(-4\)
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