Problem 13 - Algebra 2 and Trigonometry

Question

Find the value of each given expression. \(-(|-2|+|3|)\)

Step-by-Step Solution

Answer

The value of the expression \\(-(|-2|+|3|)\\) is \\(-5\\).

Step 1: Evaluate Absolute Values

The expression contains absolute values, specifically \(-|\-2| + |3|\). Remember, the absolute value of a number is its distance from zero, regardless of direction, so \(|-2| = 2\) and \(|3| = 3\).

Step 2: Simplify Inside the Parentheses

Replace the absolute values in the expression to simplify it to \(- (2 + 3)\).

Step 3: Perform the Addition Within Parentheses

Calculate \(2 + 3\) inside the parentheses to obtain \(5\). So the expression becomes \(-5\).

Step 4: Apply the Negative Sign

The expression is now \(-5\). Applying the negative sign outside, the expression remains \(-5\).

Key Concepts

Understanding Negative Numbers

Negative numbers are the numbers less than zero, and they are often denoted with a minus sign (e.g., \(-2\)).
Negative numbers represent values below zero, and they are commonly used in various fields like temperature, finance, and elevation. Let's break down how they fit into equations.

• The Role of Negative Signs: A negative sign in front of a number means it's on the opposite side of zero from its positive counterpart.
  If you see a negative sign before a number in an algebraic expression, it affects how the operation is performed.
• In Operations: When combining negative numbers with positive numbers, use their signs to determine results. For instance, \(-2 + 3 = 1\) because you move left on the number line for \(-2\) and then right three steps for \(3\).

Seeing negative signs in front of entire expressions, or within parentheses, means you should apply that sign to the result of the contained operations.

Simplification and Parentheses

Simplifying expressions with parentheses involves executing the operations within the parentheses first. This process is guided by the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Understanding parentheses in math can help you to organize expressions and control which calculations are performed first.

• Manage Simplification: In the expression \(\(-(|-2| + |3|)\)\), the operation inside the parentheses, \(2 + 3\), must be calculated before applying the negative sign. Simplifying everything inside parentheses first helps you create a cleaner expression to work with.
• Follow the Steps: Begin with what is inside the parentheses: first, evaluate any absolute values just like you have in the given problem, and then perform addition or subtraction when necessary.

This step-by-step approach ensures that you adhere to the mathematical rules that simplify operations and achieve the correct result.

Addition

Addition is one of the fundamental arithmetic operations. It's generally seen as the process of combining two or more numbers into a sum. In problems involving expressions and equations, understanding addition is essential for simplifying terms.
Let's explore how addition functions when absolute values and negative numbers are involved.

• Addition of Absolute Values: When you calculate sums using absolute values, ignore any negative signs initially. Absolute values are always non-negative, so focus on the "positive" distance from zero.
• Handling Sums: With the expression \(2 + 3\), you directly sum the two numbers, which results in \(5\). It's straightforward since both numbers are positive and the addition operator, \(+\), signifies straightforward combination.

Finally, the rule of addition with expressions helps guarantee that no matter the complexity of the terms involved, calculating correctly will yield accurate results.