Problem 101 - Prealgebra and Introductory Algebra

Question

Evaluate. $$|-y|, \text { for } y=-6$$

Step-by-Step Solution

Answer

The value of the expression |-y| for y = -6 is 6.

Step 1: Analyze the expression

We have an expression |-y|. The absolute value |x| of a real number x is its numerical value without regard to its sign. So, |-y| is the absolute value of -y.

Step 2: Substitute the given value

We are given that y = -6. Replacing y with -6 in the expression, we get |-(-6)|.

Step 3: Evaluate the absolute value

|-(-6)| equals to |6|. The absolute value of 6 is 6, because 6 is six units away from zero on the number line.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero and are represented with a minus sign (-). They are an important part of the number line which extends both to the left and to the right. Here’s what you should know about them:

• Negative numbers are on the left side of zero on the number line.
• They represent quantities that are less than zero, like temperatures below freezing or losses in finance.
• When you add a negative number, it's like subtracting its positive counterpart.
• Subtracting a negative number is like adding its positive counterpart.

Understanding how negative numbers behave can clarify many situations in math and everyday life. For example, when you encounter \(-6\), this is simply six units to the left of zero.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols like addition or multiplication. They are used to represent relationships between numbers and variables. Here are some points to clarify:

• A variable is a symbol, often a letter, that represents an unknown value or a value that can change.
• Operations in an algebraic expression must follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Expressions do not include equality signs; if they do, they become equations.

In our exercise, \-y\ is an algebraic expression where \(-y\) involves the variable \(y\) inverted to its negative value.

Substitution in Algebra

Substitution in algebra involves replacing a variable with a given numeric value in an expression or an equation. This technique helps us evaluate expressions to find their actual numerical value. Here's how substitution works:

• Identify the variable in the expression.
• Replace the variable with the given number.
• Proceed to simplify the expression using arithmetic operations.

In the provided exercise, we substitute \(y = -6\) directly into the expression \(-y\), yielding \(-(-6)\). After substitution, we simplify to find the absolute value of the result, turning our focus on evaluating \(|6|\). Knowing how to effectively substitute values is crucial for solving many math problems by transforming abstract expressions into tangible numbers.