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Chapter 3: Problem 40

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

Short Answer

Expert verified
The expression evaluates to 16.

Step by step solution

01

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.
02

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.
03

Add the Absolute Values

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 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.

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Most popular questions from this chapter

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$2(x+y)$$

Julia and Catelyn are using the Distributive Property to simplify \(3(x+2) .\) Who is correct? Explain your reasoning. $$\begin{array}{c}\text { Julia } \\\3(x+2)=3 x+2\end{array}$$ $$\begin{array}{c}\text { Catelyn } \\\3(x+2)=3 x+6\end{array}$$

Name the quadrant in which \(P(-5,-6)\) is located.

Is \(3+(x \cdot y)=(3+x) \cdot(3+y)\) a true statement? If so, explain your reasoning. If not, give a counterexample.

Divide. $$\frac{72}{-8}$$
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