Problem 53 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ |9+(-12)|+|-16| $$

Step-by-Step Solution

Answer

The result is 19.

Step 1: Simplify Inside the Absolute Values

Start by calculating the expression inside each absolute value. For the first part, compute \( 9 + (-12) \), which simplifies to \( -3 \). For the second part, the expression is just \( -16 \).

Step 2: Apply Absolute Values

Now, take the absolute value of each result from Step 1. The absolute value of \( -3 \) is \( 3 \) and the absolute value of \( -16 \) is \( 16 \).

Step 3: Add the Absolute Values

Finally, add the results from Step 2 together: \( 3 + 16 = 19 \).

Key Concepts

Simplifying Expressions

When simplifying expressions, especially those involving absolute values, it’s crucial to follow the correct steps to avoid mistakes. Absolute values can be tricky because they remove the direction of a number on a number line, leaving only its magnitude.
For example, the expression \(9 + (-12)\) needs to be calculated before considering the absolute value. Begin by simplifying the arithmetic: you are essentially adding a positive number to a negative one, which could be interpreted as subtracting in simpler terms. Here, \(9 + (-12) = -3\). Once simplified, the absolute value can be applied.
A regular step-by-step approach can aid in preventing confusion and errors.

• Simplify the expression inside the absolute value first.
• Then apply the absolute value.

Practicing this approach makes simplifying complex expressions with absolute values manageable and straightforward.

Negative Numbers

Negative numbers are numbers less than zero, often represented with a minus sign. They can initially seem daunting, especially when combined with operations like addition or subtraction. However, understanding their behavior is key to mastering arithmetic involving them.
In our original exercise, we are given a negative addition: \(9 + (-12)\). Recognize that this is the same as \(9 - 12\). Generally, adding a negative number moves the value to the left on the number line, essentially decreasing the total.
To handle these operations with ease:

• Convert addition with a negative to simple subtraction.
• Reinforce this understanding with number line visualization.

This understanding not only helps you solve absolute value problems but also builds a foundation for algebra and calculus.

Addition

When adding numbers, particularly after applying the absolute value, the process becomes more straightforward. The tricky part often happens before, when dealing with the negative numbers.
After finding the absolute value of two numbers, like in our exercise from before, you'll end up with two positive values: \(3\) and \(16\). The approach is then much like adding simple whole numbers, which is a fundamental arithmetic skill.
The following methodologies can assist:

• Break large sums into smaller chunks to simplify.
• Use a calculator for quick solutions if permitted.
• Practice mental math for improvement.

Once the numbers are rendered positive by absolute values, they should no longer complicate the addition process. Regular practice in these areas solidifies proficiency and enhances computation fluency.