Question

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

Short answer

5

Step-by-step solution

Substitute the given values

Substitute the values of \(x\) and \(y\) into the expression \( |x-y|\). Given: \(x=3\) and \(y=-2\). The expression becomes \(|3-(-2)|\).

Simplify the expression inside the absolute value

Simplify the expression inside the absolute value brackets. \(3 - (-2) = 3 + 2\). Thus, the expression becomes \(|3+2|\).

Calculate the sum

Add the numbers within the absolute value brackets. \(3 + 2 = 5\). The expression now is \(|5|\).

Evaluate the absolute value

The absolute value of 5 is 5. Thus, \(|5| = 5\)

Key concepts

Substitution in Algebra

Substitution in algebra is an essential technique where we replace variables with given values. Imagine a variable like a placeholder or a box you can fill with any specified number.
In our example, the exercise provides specific values for the variables:

• Given:
  \(x = 3\)
  \(y = -2\)
  

We use these replacement values to transform the algebraic expression into a simpler numerical one. This substitution allows us to convert abstract expressions into calculations we can handle. By plugging in the provided values, we take the first step towards solving the problem. It's much like swapping out variables for their actual value in the problem at hand.

Evaluating Expressions

Once we substitute the values into the expression, the next step is to evaluate it. Evaluating expressions involves carrying out the necessary mathematical operations to simplify them. For example, in our given problem, after substituting, we get:

• \(| 3 - (-2) |\)

Here, we need to recognize the double negative. Notice that subtracting a negative number is the same as adding the positive counterpart. This changes our expression to:

• \(| 3 + 2 |\)

Evaluating expressions requires careful attention to the operations involved, ensuring we handle each step correctly: addition, subtraction, multiplication, and division as dictated by the problem.

Simplifying Expressions

After evaluating the expression, the next critical step is simplification. Simplifying means reducing the expression to its simplest form. Here, we carry out straightforward arithmetic operations. In our exercise, after recognizing that subtracting a negative is like adding a positive, we get:

• \( 3 + 2\)

We then simplify by performing this addition to get:

• \( 5\)

Simplification helps in reducing complexity and making the expression easier to grasp, or further work with in the problem.

Absolute Value

Absolute value is a measure of how far a number is from zero, regardless of direction on the number line. For our expression, once simplified to \(|5|\), the final step is evaluating this absolute value.
Absolute value is represented by two vertical lines on either side of the number or expression. The word 'absolute' means the value is 'stringent' or always positive.

• For any positive number or zero, the absolute value is the number itself: \(|5| = 5\)
• For any negative number -a, the absolute value is the positive version of that number: \(|-5| = 5\)

So, in our problem, the expression resolves to 5:
\( |5| = 5 \)
Understanding absolute value is crucial since it depicts the magnitude of a number detached from its sign.