Problem 63 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -62-71-(-37)+99 $$

Step-by-Step Solution

Answer

The result of the operation is 3.

Step 1: Simplify Double Negation

The expression given is \(-62-71-(-37)+99\). Begin by addressing the double negation: Subtracting a negative number is equivalent to adding a positive number. Therefore, change \(-(-37)\) to \(+37\). The expression becomes:\(-62-71+37+99\).

Step 2: Group Negative Numbers

Focus first on the negative numbers: \(-62\) and \(-71\). Combine them: \(-62 - 71 = -133\).

Step 3: Add Positive Number

Include the positive number from the simplification: \(-133 + 37\). Calculate:\(-133 + 37 = -96\).

Step 4: Add the Second Positive Number

Finally, add the remaining positive number: \(-96 + 99\). Calculate:\(-96 + 99 = 3\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are typically represented with a minus sign in front, like

• \(-1, -2, -3\), and so on.
• These numbers appear to the left of zero on a number line.

Negative numbers are important in real life, representing things like debt or temperature below zero. When you have one negative number, its value is less than zero, making it the opposite of positive numbers.
When dealing with operations involving negative numbers, it's essential to remember:

• Subtracting a positive number makes the result more negative.
• Adding a positive number will result in a value closer to zero.

Understanding how negative numbers work is crucial for performing arithmetic operations correctly.

Demystifying Double Negation

Double negation is a situation where two negative signs appear, one after the other. In arithmetic, negating a negative number results in a positive number.
For example,

• Subtraction of a negative number is the same as addition.
• If you see \(-(-x)\), it's the same as \(+x\).

In the given problem, you encountered the expression \(-(-37)\). By transforming it:\(+37\), you are effectively adding 37 instead of subtracting. Think of it like taking away a minus sign, which switches the number to a positive.
This transformation is crucial when simplifying expressions as it helps resolve potential confusion and simplifies further calculations.

The Basics of Addition of Integers

Addition of integers involves combining numbers to get a sum. In simple terms:

• If both numbers are positive, you just add them.
• If one is negative and one is positive, you subtract the smaller absolute value from the larger.
• With two negative numbers, add their absolute values and keep the negative sign.

Steps can vary, especially as seen in our problem:1. The expression first simplifies the double negation to \(-62-71+37+99\).2. Combining two negative numbers \(-62\) and \(-71\) yields another negative number.3. Adding the results progressively, starting with positive values, makes it simple to handle.
Ultimately, the correct approach to adding integers includes recognizing signs and understanding how positive and negative numbers interact. With practice, identifying these patterns helps in solving complex arithmetic expressions efficiently.