Problem 98 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. In each exercise, write an integer that is the result of the given situation. You lose \(\$ 50\) and then you misplace \(\$ 10\).

Step-by-Step Solution

Answer

The result of the given situation is -60

Step 1: Identify the initial loss

The first step is to identify the initial loss of money due to losing. This is given to be \(\$ 50\). To represent this as an integer, we simply use -50. This is because when money is lost, it's a subtraction or negative impact to the total amount of money you have. So -50 represent a loss of \(\$ 50\).

Step 2: Identify the subsequent loss

The second step is to identify the subsequent loss of money, due to misplacing \(\$ 10\). Again, this is a loss so it will be represented as a negative integer. Hence this loss is represented by -10.

Step 3: Combine both losses

The final step is to combine both losses together. This is done by adding the integer value of the two losses together. Remember when you add two negative numbers, you get a more negative number (as you're going further into 'debt'). Adding -50 and -10 gives \(-50 - 10 = -60\).

Key Concepts

Integer Operations

Understanding integer operations is crucial as they form the foundation of algebra and many other mathematical concepts. In essence, integer operations involve performing basic arithmetic such as addition, subtraction, multiplication, and division on integers. An integer, in simple terms, is a whole number that can be positive, negative, or zero.

When dealing with integers, it's important to remember:

• Negative numbers are denoted by a minus sign (e.g., -3).
• Positive integers can be written without a '+' sign (e.g., +5 or simply 5).
• Zero is neutral, meaning it is neither negative nor positive.

Performing operations on these kinds of numbers follows specific rules that help us to consistently calculate correct results.

Negative Numbers

Negative numbers represent quantities less than zero, often described as a loss, debt, or below a baseline amount. They are essential in real-life scenarios, like financial calculations and temperatures below freezing.

Key points about negative numbers include:

• They are written with a minus sign (-) before the number (e.g., -50).
• When visualized on a number line, negative numbers appear to the left of zero.

This concept is pivotal in understanding the effects of subtracting or adding negative integers, as demonstrated in our exercise. For instance, losing money can be seen as a journey to deeper negatives on a number line.

Addition of Integers

Adding integers can sometimes be confusing, especially when dealing with negative numbers. However, some simple rules can make this process straightforward:

• Adding two positive numbers is like regular addition.
• Adding a positive and a negative number requires you to subtract the smaller from the larger number, and the result takes the sign of the larger absolute value.
• Adding two negative numbers results in a more negative number. This is similar to moving further away from zero in the negative direction.

In the exercise, when we combined -50 and -10, both were negative, leading us to a sum of -60. It's like digging deeper into a hole; the more you dig, the further from the surface you get.