Problem 70 - Prealgebra

Question

Simplify each of the following. $$-|-3|$$

Step-by-Step Solution

Answer

The simplified expression is \(-3\).

Step 1: Understand the Absolute Value

The expression \(-|-3|\) involves the absolute value of a number. Absolute value refers to the distance a number is from zero on the number line, disregarding whether it is negative or positive. Thus, the absolute value of \(-3\) is \(|-3| = 3\).

Step 2: Recognize the Negative Sign

Once the absolute value is determined as \(3\), consider the negative sign outside the absolute value bars. The expression now transforms into \(-3\) since the negative sign indicates the negation of the absolute value result.

Step 3: Simplify the Expression

Now, simply express the calculation: \(-|-3| = -3\). This is the simplified form of the original expression.

Key Concepts

Integer Operations

Integer operations are the basic mathematical procedures involving whole numbers, which include addition, subtraction, multiplication, and division. When working with integers, it's important to pay attention to the sign of each number. This is because the sign determines the direction on the number line. Here are some key pointers about integer operations:

• Addition of integers: If the signs are the same, add the absolute values and keep the sign. For example, o\(3 + 5 = 8\), and a negative example: o\((-3) + (-5) = -8\).
• Subtraction of integers: Subtracting is the same as adding the opposite. So, o\(6 - 2\) becomes o\(6 + (-2) = 4\).
• Multiplication and division of integers: If the signs are the same, the result is positive. If the signs differ, the result is negative. For example, o\((-3) \times 4 = -12\), but o\((-3) \times (-4) = 12\).

Understanding these operations helps in performing calculations swiftly and accurately, especially in expressions involving both positive and negative integers.

Negative Numbers

Negative numbers are less than zero and they appear on the left side of zero on a number line. In mathematical expressions, they are usually preceded by a minus sign. Knowing how to handle negative numbers is crucial in both simple and complex mathematical problems. Here’s what to keep in mind:

• Subtracting a negative number: Subtracting a negative is the same as adding a positive. For example, o\(5 - (-3) = 5 + 3 = 8\).
• Adding a negative number: This is equivalent to simple subtraction. For instance, o\(7 + (-2) = 7 - 2 = 5\).
• Multiplication and division with negative numbers: If you're dealing with a negative number, switching the signs of your result as shown in integer operations is essential. Always consider how negative signs will affect your end result. o\((-2) \times 4 = -8\), o\(8 \div (-4) = -2\).

Negative numbers are fundamental in understanding financial debts, temperature changes below zero, and various other real-world situations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to understand and work with. During simplification, it’s key to remember certain mathematical properties and rules. Here's a breakdown of how to approach simplifying expressions:

• Identify any absolute value: Absolute values convert any number within the bars to its non-negative equivalent, which simplifies calculations influenced by negatives.
• Combine like terms: Terms with the same variable and exponent can be combined by adding or subtracting their coefficients. For instance, o\(3x + 5x = 8x\).
• Use distribution and factoring: Apply the distributive property when needed and factor out common terms to achieve a simplified expression. For example, to simplify o\(3(x + 4) = 3x + 12\).
• Apply arithmetic operations: Follow order of operations (PEMDAS/BODMAS) to solve or simplify expressions systematically. For instance, simplifying o\(-|-3|\) involves finding the absolute value o\(|-3| = 3\) and applying the negative sign to get o\(-3\).

Simplifying helps solve equations and provides a clearer understanding of complex mathematical problems. With practice, anyone can master simplifying expressions without difficulty.