Problem 62 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ |-3|-(-6) $$

Step-by-Step Solution

Answer

Insert \(a>\) to make the statement true.

Step 1: Evaluate the Absolute Value

Evaluate the absolute value of each number in the expression. The absolute value of \(-3\) is \(|-3| = 3\). Absolute values turn all negatives into positives. So, our expression is equivalent to \(3 - (-6)\).

Step 2: Simplify the Subtraction of a Negative

Simplify the subtraction of a negative number. Subtracting a negative is the same as adding its positive. Therefore, \(3 - (-6)\) is the same as \(3 + 6\).

Step 3: Calculate the Simplified Expression

Add the numbers together. The expression \(3 + 6\) calculates to \(9\). So, we have \(9\).

Step 4: Decide the Comparison Operator

Insert the inequality \(a<\) or \(a>\) such that \(|-3| - (-6)\) holds true as a statement. Since \(9\) is greater than \(0\), the symbol that makes the statement true is \(>\).

Key Concepts

Understanding Inequality Symbols

Inequality symbols are crucial in mathematics for comparing values. These symbols help us identify which of two numbers or expressions is larger or smaller.

The two primary inequality symbols are:

• "<" (less than) – indicates that the number on the left is smaller than the number on the right.
• ">" (greater than) – indicates that the number on the left is larger than the number on the right.

When evaluating expressions with absolute values, we use these symbols to compare the evaluated numbers. It's important to understand what each symbol means so you can select the correct one based on the numbers you're comparing. In the context of our original exercise, once we've simplified to find that the result is 9, the symbol "greater than" ( ">" ) clearly describes the relationship between our calculated result and zero.

Handling Negative Numbers

Negative numbers might seem tricky, but they're simply numbers less than zero. On a number line, they appear to the left of zero.

When dealing with negative numbers, especially in expressions involving subtraction or addition:

• Subtracting a negative number is the same as adding the positive equivalent. For example, subtracting \(-6\) is the same as adding 6, as shown in our expression step-by-step solution.
• Adding a negative number is akin to subtracting the absolute value of that number. For instance, adding \(-3\) would be like subtracting 3 from the positive number.

Understanding these interactions is key to effectively simplifying expressions and solving equations, particularly when they involve multiple negative values.

Simplifying Expressions with Absolute Value

Simplifying expressions is about reducing them to their most basic form. Absolute value, in this regard, plays a significant role because it allows us to evaluate and convert negative numbers into positive numbers.

The absolute value of a number is its distance from zero on the number line, regardless of direction. Here’s how you simplify with absolute values:

• Convert negative numbers within the absolute value symbols to their positive counterparts. For example, \(|-3|\) becomes 3.
• Perform arithmetic operations after evaluating absolute values. Following our previous exercise, once we calculated \(|-3|-(-6)\) as \(3 - (-6)\), we proceed by changing subtraction to addition, making it \(3 + 6\).

The result is a straightforward calculation that reveals the final number you need to compare, utilizing inequality symbols. Mastering these steps simplifies more complex mathematical expressions and ensures accurate calculations.