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Chapter 0: Problem 52

In Problems 49-58, find the value of each expression if \(x=3\) and \(y=-2\). \(|x|-|y|\)

Short Answer

Expert verified
The value is 1.

Step by step solution

01

Substitute the Values

First, substitute the given values of the variables into the expression. Replace \( x \) with 3 and \( y \) with -2 in the expression \( |x| - |y| \).
02

Evaluate the Absolute Values

Find the absolute values of \(x\) and \(y\). \( |3| = 3 \) since the absolute value of 3 is 3, and \( |-2| = 2 \) since the absolute value of -2 is 2.
03

Subtract the Absolute Values

Subtract the absolute value of \(y\) from the absolute value of \(x\): \( 3 - 2 = 1 \).

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Key Concepts

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

absolute value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number.
For example:
  • The absolute value of 3 is 3, because it is 3 units away from zero.
  • The absolute value of -2 is 2, because it is also 2 units away from zero.

Absolute values are denoted by vertical bars. For instance, \(|x|\) represents the absolute value of x.
This concept is crucial when solving expressions that involve differences between absolute values, such as \(|x| - |y|\).
substitution
Substitution is a method where we replace variables in an expression with their given values. This step simplifies the problem by allowing us to work with known numbers instead of variables.
For instance:
  • In the expression \( |x| - |y| \), if x = 3 and y = -2, we substitute these values into the expression.
  • This gives us \( |3| - |-2| \).
  • We then proceed to evaluate the absolute values.

Substitution is often the first step in problem-solving as it sets the stage for further calculations.
evaluating expressions
Evaluating expressions involves performing the operations in the expression to find its value. This process typically follows a series of steps:
  • First, substitute the given values of the variables.
  • Next, compute any absolute values.
  • Finally, perform the arithmetic operations as specified in the expression.

In our example, after substituting and evaluating the absolute values, we subtract them: \( |3| - |-2| = 3 - 2 = 1 \).
This final step gives us the value of the expression.

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Most popular questions from this chapter

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{3}-1}{2 \sqrt{3}+3}$$

Simplify each expression. $$25^{3 / 2}$$

Use a calculator to approximate each radical. Round your answer to two decimal places. $$\sqrt[3]{-5}$$

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{(x+4)^{1 / 2}-2 x(x+4)^{-1 / 2}}{x+4} \quad x>-4$$

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt{9 x^{5}}$$
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