Problem 46 - Algebra 1

Question

Evaluate the expression. $$-4(|y-12|) \text { when } y=5$$

Step-by-Step Solution

Answer

-28

Step 1: Understand the Absolute Value

The absolute value of 'a' denoted by '|a|', is the distance of 'a' from zero on the number line, regardless of the direction. For instance, if 'a' is a positive number or zero, '|a|' is 'a'. If 'a' is a negative number, '|a|' is '-a'.

Step 2: Compute the Absolute Value

Compute the absolute value of 'y-12' by substituting the value of 'y' which is '5'. Hence, '|5-12|' equals to '|-7|' which is '7'.

Step 3: Evaluate the Expression

Evaluate the expression '-4(|y-12|)' by multiplying '-4' and the computed absolute value in step 2 which is '7'. Hence, '-4' multiplied by '7' equals to '-28'.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols like addition, subtraction, multiplication, and division. They form the foundation of algebra, allowing us to create mathematical phrases that represent real-world situations. For example, the expression

• \(-4(|y-12|)\)

combines parentheses, an absolute value, and a variable.
Understanding algebraic expressions involves recognizing each component. Constants are fixed numbers (like -4 and 12), while variables (such as \(y\)) stand for numbers that can vary. Operators like addition and multiplication connect these terms, and certain expressions use functions like absolute value to change how terms are calculated. With proficiency in algebraic expressions, students can solve a wide range of problems by accurately computing or simplifying these expressions.

Evaluating Expressions

Evaluating expressions is the process of finding the value of an algebraic expression by performing the operations. This often requires knowing the exact values for any variables in the expression.
Consider the expression from the exercise:

• \(-4(|y-12|)\)

Once we know that \(y=5\), we can replace the variable with this number.
We first address the absolute value, because operations within parentheses or absolute value bars should be resolved first. Then, we compute the expression inside the absolute value, yielding \(|5 - 12| = |-7|\), which simplifies to 7. Finally, by proceeding with the multiplication,

• \(-4 \times 7 = -28\)

This gives us the final evaluated result. Evaluating expressions requires careful attention to the order of operations, particularly when dealing with absolute values and nested operations.

Substituting Values

Substituting values involves replacing the variables in an algebraic expression with known numbers. This step is essential for evaluating expressions because it allows numbers to substitute for variables, turning an algebraic phrase into a numeric one that is easier to calculate.
Taking the expression \(-4(|y-12|)\) once more, the substitution step takes place when we take the given \(y=5\) and replace \(y\) in the expression.

• This changes the equation to \(-4(|5-12|)\)

which can then be handled using arithmetic operations. Substituting values is a straightforward process but requires accuracy to ensure that the rest of the evaluation is correct. Any mistakes in substitution can lead to incorrect results, so it is important to carefully follow the steps, substituting the numbers at the appropriate places with precision.