Question

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

Short answer

The value is 1.

Step-by-step solution

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 \( |x| - |y| \).

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.

Subtract the Absolute Values

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

Key concepts

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.