Question

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

Short answer

1

Step-by-step solution

- Substitute Values

Replace the variables with their given values. Here, substitute 3 for x and -2 for y in the expression \( |x+y| \). This gives us \( |3 + (-2)| \).

- Simplify Inside the Absolute Value

Perform the arithmetic operation inside the absolute value. Therefore, \( 3 + (-2) = 1 \). The expression now becomes \(|1|\).

- Evaluate the Absolute Value

The absolute value of 1 is 1, because the absolute value measures the distance from zero and is always non-negative. Therefore, \(|1| = 1\).

Key concepts

Substitution

In this exercise, we need to find the value of an expression given specific values for the variables involved. The first step in solving the problem is substitution.
In our case, we have the variables \(x\) and \(y\), and we're given that \(x = 3\) and \(y = -2\).
Substitution means replacing these variables with their given values in the expression.
This is a common technique in algebra used to make the expression easier to work with.
By substituting \(x = 3\) and \(y = -2\) into the expression \(|x + y|\), it becomes \(|3 + (-2)|\).

Simplification

Once we substitute the values, our next task is simplification. This step helps to reduce the complexity of the expression.
In our case, we need to simplify \(3 + (-2)\) inside the absolute value.
Remember that adding a negative number is the same as subtracting the positive counterpart.
Therefore, \(3 + (-2)\) simplifies to \(1\).
After simplification, the expression now reads \(|1|\).
Simplification makes the expression much easier to evaluate and understand.

Arithmetic Operations

The final concept we deal with in this exercise is handling arithmetic operations within absolute value expressions.
Arithmetic operations like addition, subtraction, multiplication, and division follow standard rules, even within absolute values.
The absolute value function, denoted by vertical bars \(| |\), measures how far a number is from zero on the number line, ignoring any negative sign.
Therefore, \(|1| = 1\) because 1 is one unit away from zero.
The result of this arithmetic operation confirms that the absolute value of 1 is indeed 1.
Through these steps, we see that clear substitution, simplification, and accurate arithmetic operations lead to the correct result.