Problem 65 - College Algebra Essentials

Question

Evaluate each algebraic expression for x = 2 and y = -5. $$\frac{y}{|y|}$$

Step-by-Step Solution

Answer

-1

Step 1: Compute the absolute value of y

The absolute value of -5 is 5, since absolute value makes any negative number positive and leaves any positive number or zero as it is. Therefore, |y| = |-5| = 5.

Step 2: Perform the division

Replace y with its value -5 and |y| with its value from step 1. That is, compute the expression \(-5 / 5\) which equals to -1. Hence, the expression simplifies to -1.

Key Concepts

Absolute Value

The absolute value of a number represents its distance from zero on a number line, without considering direction. Thus, absolute value is always non-negative. When dealing with algebraic expressions, the absolute value can simplify computations, especially when working with variables that may take on negative or positive values.

To calculate the absolute value of a number, you merely strip away any negative sign. For example, the absolute value of \-5 is 5, because you ignore the negative and consider only how far 5 is from zero.

• If the number is positive or zero, the absolute value is the number itself: \( |7| = 7 \) and \( |0| = 0 \).
• If the number is negative, the absolute value is the opposite of the number: \( |-3| = 3 \).

Whenever solving expressions involving absolute values, always determine the absolute value first before executing further operations like addition, subtraction, multiplication, or division.

Substitution

Substitution in algebra simplifies expressions by replacing variables with given numerical values. This process helps derive a specific outcome from a general expression. When asked to evaluate an expression for certain values of variables, substitution is the method.

In the exercise, the variable \("y"\) is substituted with the value \(-5\). This transforms the given algebraic expression into a direct calculation. Follow these steps:

1. Identify the variable in the expression.2. Replace the variable with its corresponding value.3. Proceed with the calculation using the substituted values.

• For instance, replacing \( y \) with \(-5\) turns the expression \( \frac{y}{|y|} \) into \( \frac{-5}{|-5|} \).

Substitution is essential in simplifying expressions and verifying solutions.

Division in Algebra

Division in algebra involves dividing one number, or expression, by another. When working with algebraic expressions, division can simplify or reduce expressions into more digestible parts. Correctly handling signs and values is crucial.

Division is represented by a fraction bar in algebraic expressions. For instance, in \( \frac{y}{|y|} \), you're dividing "y" by its absolute value \( |y| \). The process follows these steps:

1. Calculate the dividend — the number you are dividing.2. Determine the divisor — the number you are dividing by.

• Apply the equation: \( \frac{-5}{5} \), resulting in \(-1\) since the negative sign remains.

Remember, dividing by a number is multiplying by its reciprocal. In algebra, ensuring that both numbers—typically variables—align correctly is key to a valid solution.