Problem 28 - Elementary Algebra

Question

Determine each of the values, \(-|-31|\)

Step-by-Step Solution

Answer

Prompt: As a teacher, create a short answer based on the following step by step solution: Question: Evaluate the expression -|-31|. Answer: The given expression is -|-31|. First, we find the absolute value of -31 which is 31. Then, we take the negative of 31, resulting in -31. So the expression -|-31| evaluates to -31.

Step 1: Calculate the absolute value of -31

To find the absolute value of a number, we simply take the non-negative value of the number. In this case, the absolute value of \(-31\) is 31, since 31 is the non-negative counterpart. So, \(|-31| = 31\).

Step 2: Take the negative of the absolute value

Now that we have the absolute value of \(-31\), we need to take the negative of that value. So, the expression \(-|-31|\) becomes \(-31\). Thus, the final answer for the given expression is \(-31\).

Key Concepts

Negative Numbers

Negative numbers represent values that are less than zero. These numbers are often visualized on a number line to the left of zero. Working with negative numbers involves understanding how they behave in calculations. For example, when you add a negative number, it's like subtracting a positive number, and when you subtract a negative number, it's similar to adding a positive number.

Understanding negative numbers is essential in various math problems, especially in algebra where they frequently appear in equations and expressions. They can also impact computations such as multiplication and division, where multiplying two negative numbers results in a positive product.

• Negative numbers are represented with a minus sign (-).
• On a number line, they appear to the left of zero.
• They are crucial in many fields such as finance for denoting debts or losses.

Expressions

Expressions in mathematics are combinations of numbers, variables, and operations. They represent values and can be simplified or evaluated to find those values. For instance, in our example problem, the expression \(-|-31|\) contains an operation, the absolute value, which is crucial in finding the result.

Part of working with expressions is knowing how to apply various algebraic concepts, such as simplifying or evaluating them. Expressions can be numerical, involving only numbers and operations, or algebraic, which include variables.

• Expressions do not have equal signs (unless they are equations).
• The goal is often to simplify or find the value of an expression.
• They are foundational in algebra, calculus, and more advanced mathematics.

Algebraic Concepts

Algebraic concepts expand on simple arithmetic by incorporating variables, constants, and more complex structures like equations and inequalities. In the given problem, the concept of absolute value, indicated by the vertical bars \(|\cdot|\), is a simple yet profound algebraic concept.

Absolute value is the distance of a number from zero on the number line, always expressed as a non-negative number. It transforms negative numbers to their positive counterparts, which is why in the example, \(|-31|\) becomes 31. Further algebraic manipulation can then apply, such as negating this absolute value to achieve the final expression \(-31\).

• Algebra introduces symbols and letters (variables) to represent numbers.
• Understanding operations like absolute value is key in dealing with expressions effectively.
• These concepts form the backbone of solving and understanding complex equations in higher mathematics.