Problem 40 - Prealgebra

Question

Evaluate each expression if \(x=-12, y=4,\) and \(z=-1\) $$|x|+|y|$$

Step-by-Step Solution

Answer

The expression evaluates to 16.

Step 1: Understand the Absolute Value

The absolute value of a number is its distance from zero on a number line, always expressed as a non-negative value. Thus, \(|x|\) means the distance of \(x\) from zero.

Step 2: Evaluate the Absolute Values

Given \(x = -12\) and \(y = 4\), we need to find \(|x|\) and \(|y|\). Calculate: \(|x| = |-12| = 12\) and \(|y| = |4| = 4\). Both values are positive as they represent distances.

Step 3: Add the Absolute Values

Now, we add the absolute values calculated previously: \(|x| + |y| = 12 + 4 = 16\).

Key Concepts

Evaluating Expressions

Evaluating expressions means calculating the value of an expression by substituting variables with their given numbers. In the exercise, you're asked to substitute \(x = -12\) and \(y = 4\) into the expression \(|x| + |y|\). Here’s how to systematically approach such problems:

• First, identify the variables and their corresponding values from the problem statement.
• Next, replace each variable in the expression with these actual numbers.
• Finally, perform the necessary arithmetic operations to calculate the final result.

This step-by-step strategy ensures you don't miss any details and allows you to solve the expression accurately. Remember, the order of operations (PEMDAS/BODMAS) is crucial when dealing with complex expressions.

Integer Operations

Integer operations involve basic arithmetic operations with whole numbers, including positive and negative values. It’s essential to handle integers correctly, especially when they are inside absolute value expressions. Here are some key points:

• Addition: When adding integers, same signs mean you add and keep the sign, but if they have different signs, subtract and use the sign of the larger number.
• Subtraction: This is the same as adding the opposite. So, \(5 - 12\) is the same as \(5 + (-12)\).
• Absolute Values: Remember, absolute values convert negative numbers to positive, which is crucial in our exercise. For example, \(|-12|\) becomes \(12\).

These rules help simplify problems, ensuring you perform the operations correctly regardless of the integers' nature.

Number Line Distance

The absolute value of a number represents its distance from zero on a number line. This concept is fundamental in understanding how to evaluate expressions involving absolute values.

• Distance is always a non-negative value, which explains why the absolute value of any number is positive.
• For example, the distance of \(-12\) from zero is \(12\), so \(|-12| = 12\).
• Similarly, \(4\) is \(4\) units away from zero, hence \(|4| = 4\).

Conceptualizing numbers on a number line helps you visualize these distances, making it easier to grasp why absolute values behave as they do. It's a neat way to appreciate the 'space' numbers occupy in this visual model.