Problem 19 - Intermediate Algebra

Question

Perform the operations. See Example 1 . $$ -9+(-8)+4 $$

Step-by-Step Solution

Answer

The result is -13.

Step 1: Simplify the Expression

We begin by simplifying the given mathematical expression: \(-9 + (-8) + 4\).This involves handling addition and subtraction of integers.

Step 2: Add the First Two Numbers

First, let's add the first two numbers together. Combine \(-9\) and \(-8\): \(-9 + (-8) = -17\).

Step 3: Add the Result with the Third Number

Now, add the result from Step 2 to the third number. Combine \(-17\) and \(4\): \(-17 + 4 = -13\).

Step 4: Conclusion: Final Result

The simplified result of the operation \(-9 + (-8) + 4\) is \(-13\).

Key Concepts

Addition of Integers

When dealing with integer operations, it's essential to understand how to add integers, especially when negative numbers are involved. Addition is combining values to find their total. With integers, it's crucial to pay attention to their signs (positive or negative).

• Same Sign: When adding integers with the same sign, such as (-9) + (-8), you add their absolute values together and keep the common sign. Here, 9 plus 8 is 17, and since both numbers are negative, the answer is (-17).
• Different Sign: When adding numbers with different signs, instead of adding, you subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value. If you have a positive 4 with the negative sum (-17) from our exercise, \(|17| - |4| = 13\), and since 17 is higher in absolute terms and is negative, we have (-13).

This approach makes sense since positive numbers can cancel out negative numbers to some degree. Understanding this helps make the addition of integers easier to compute.

Subtraction of Integers

Subtraction of integers can be thought of as adding the opposite. When you see a subtraction problem with integers, convert it into an addition problem by changing the sign of the number that’s being subtracted.
This way, the rules of addition apply, making the task easier. For example, if you begin with adding (-9 + (-8) + 4), but encounter a subtraction, such as (-9) - 8, transform it
into (-9) + (-8). Now, follow the same process as adding integers with the same sign. Here's a basic guideline:

• Rewrite: Always rewrite subtraction as adding the opposite number. For instance, 5 - 8 becomes 5 + (-8).
• Addition Rules: Use the addition rules for integers. Same signs get added, different signs will subtract their absolute values.

Understanding subtraction in terms of addition simplifies your process and helps prevent errors in operations.

Simplifying Expressions

Simplifying expressions involves combining like terms and performing operations step by step to reduce them to a simpler form, which is especially crucial in algebra. This not only makes the expressions manageable but also helps in accurately determining their values.

• Step-by-Step Approach: Tackle expressions methodically by prioritizing operations following standard math rules like parentheses, exponents, multiplication/division, and addition/subtraction (PEMDAS).
• Combine Simplified Steps: As in the original exercise, start with simple additions or subtractions and gradually simplify. First, resolve (-9 + (-8)), which simplifies to (-17), and then add the 4, leading to a final simplified answer of (-13).

This structured breakdown aids in tracking and verifying each stage of calculation so numbers are handled progressively and systematically. Not only does this prevent mistakes, but it also provides clarity for complex problem-solving.