Problem 42 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -9+15+(-5) $$

Step-by-Step Solution

Answer

The result of the expression is 1.

Step 1: Rewrite the Expression

First, let's rewrite the expression so that it is easier to understand. We have\[-9 + 15 + (-5)\]This expression contains both positive and negative integers.

Step 2: Group the Negative Numbers

Now, group all the negative numbers together. In this expression, the negative numbers are \[-9\]and \[-5\]. So, add them together first:\[-9 + (-5) = -14\].

Step 3: Add the Positive Number

Next, take the result from Step 2, which is \(-14\), and add the positive number, \(15\):\[15 + (-14) = 1\].

Step 4: Simplify to Get the Final Answer

The simplified expression \(15 + (-14)\) gives us \(1\). Therefore, the final result of the expression \(-9 + 15 + (-5)\) is \(1\).

Key Concepts

Understanding Negative Numbers

Negative numbers can often seem tricky at first, but with a little practice, they become much easier to handle. A negative number is any number less than zero, appearing with a minus sign (-) before it. They are the opposite of positive numbers and are found below zero on a number line.
For instance, -9 is a negative number because it is 9 units below zero. When working with negative numbers in arithmetic, there are a few key points to remember:

• Subtracting a negative number is the same as adding its positive counterpart. For example, \(-(-5)\) is equivalent to adding \(5\).
• When adding two negative numbers together, you add their absolute values and then attach a negative sign to the result.
• Similarly, adding a negative number to a positive number is like subtraction. You subtract the smaller absolute value from the larger one, and the sign of the result depends on which absolute value was larger.

Understanding these rules can help you handle negative numbers with ease in any expression.

The Power of Grouping Integers

Grouping is a powerful tool to simplify arithmetic expressions, especially when dealing with both positive and negative integers. When you group integers, you are effectively organizing the numbers in a way that makes computation easier to follow.
In our expression, \(-9 + 15 + (-5)\), grouping the negative integers first helps you focus on a smaller part of the problem. By adding the negative numbers \(-9\) and \(-5\) together, you simplify that part to \(-14\).
This strategy is beneficial because:

• It reduces the number of operations you need to perform at once.
• Makes it clearer to see what your next step should be, by simplifying the expression incrementally.
• Helps to avoid errors by dealing with smaller, more manageable portions of the problem.

Grouping integers is especially handy when dealing with larger sets of numbers or more complex problems, as it breaks them down into simpler parts.

Simplifying Expressions Step by Step

Simplifying expressions involves breaking down an arithmetic problem into the easiest, most understandable form. This process is all about systematically reducing complexity by performing basic operations one step at a time.
Consider the original problem \(-9 + 15 + (-5)\):

• Start Small: Begin by grouping similar terms, like the negative numbers in our example. This simplifies the task of addition or subtraction.
• Perform Arithmetic: Next, take the grouped numbers and perform their respective operations, as we did with \(-9 + (-5) = -14\).
• Combine Results: Finally, combine this result with the remaining numbers, like adding \(15\) to \(-14\), to achieve a final, simplified expression.

Each of these steps is about making the expression simpler, turning the problem from a potentially confusing equation into a straightforward calculation. As you practice, this process will become second nature, and simplifying any arithmetic expression will be a breeze!