Problem 66 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<,>,\) or \(=\) in the shaded area to make a true statement. $$[(-8)+(-6)]+10\quad\square\quad-8+[9+(-2)]$$

Step-by-Step Solution

Answer

The correct symbol to insert is <, so the statement becomes \[ [(-8)+(-6)]+10 < -8+[9+(-2)] \].

Step 1: Simplify the Left Side

In the expression \((-8)+(-6)+10\), we first add together -8 and -6 to get -14. When you add -14 and +10, the result is -4.

Step 2: Simplify the Right Side

In the expression \(-8+(9+(-2))\), we first add together 9 and -2 to get 7. Then we add 7 to -8 to get -1.

Step 3: Compare Both Sides

Comparing -4 (left side) and -1 (right side), we find -4 is less than -1.

Key Concepts

Simplifying Expressions

One of the fundamentals of algebra is learning how to simplify expressions. Simplification makes complex problems more manageable by breaking them down into simpler parts that are easier to deal with. In the given exercise, simplification involves adding and subtracting integers. For instance, simplifying the expression \( (-8) + (-6) + 10 \) requires combining like terms, which in this case are the integers. The process of addition and subtraction is applied to these numbers in a particular sequence, often following the associative property that allows us to group numbers in any way we find convenient.

Simplification is not merely about making the expression shorter; it involves understanding the relationships between numbers and how they combine. When numbers are combined thoughtfully, it helps to unveil the true value of the expression. As demonstrated in the solution, simplifying each side of the inequality separately allows for an easier comparison between the two, highlighting the importance of this crucial step in solving algebraic problems.

Comparing Integers

Comparing integers is a basic yet essential skill in algebra and general mathematics. It requires an understanding of the number line and the relative placement of numbers in regards to zero—the central point which divides positive and negative integers. In our exercise, we are asked to compare \( -4 \) and \( -1 \) to determine which is larger or smaller.

Visualizing or mentally mapping these numbers on a number line can be a helpful approach. It's important to remember that the further left a number is on the number line, the smaller its value. Therefore, despite both \( -4 \) and \( -1 \) being negative, \( -4 \) is less than \( -1 \) since it is farther away from zero on the left side. Recognizing these patterns and internalizing the concept of the number line is crucial for effectively comparing integers and understanding their real-world implications.

Algebraic Operations

Algebraic operations form the backbone of algebra, encompassing addition, subtraction, multiplication, and division. The step-by-step solution to the given problem shows the use of algebraic operations in simplifying and solving expressions. These operations follow specific properties and rules that govern how numbers and variables interact.

For example, the associative property used in algebraic operations, allows for the regrouping of numbers when adding or subtracting, without changing the result. This property is particularly useful when simplifying expressions as it provides flexibility in the calculation process. Consistent practice in applying these operations can lead to a more intuitive understanding of algebra and significantly improve problem-solving skills. It's through these fundamental operations that we can manipulate and solve not just simple equations, but complex algebraic expressions that model real-life scenarios.