Problem 61 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ -(-5) \quad-10 $$

Step-by-Step Solution

Answer

Insert the symbol '>': \(-(-5) > -10\).

Step 1: Evaluate the Expression

First, we need to evaluate the expression on the left side of the inequality: \ \(-(-5)\) simplifies to \(5\). This is because the negative of a negative number is positive.

Step 2: Compare both Values

Now that we have the value from the left side, \(5\), we compare it to the right side, \(-10\). \ We know that positive numbers are always greater than negative numbers.

Step 3: Determine the Correct Symbol

Since \(5\) is greater than \(-10\), the correct symbol to insert into the inequality is \(>\). Thus, the true statement is \(-(-5) > -10\).

Key Concepts

Negative Numbers

Negative numbers can sometimes seem a bit tricky, but once you understand the basics, they become a lot easier to handle. In mathematics, negative numbers are numbers that are less than zero. They are typically represented by a minus sign (-) in front of a number.
For example:

• -1, -2, -3 are all negative numbers.
• They are the opposites of positive numbers.

When dealing with expressions that involve negative numbers, it is essential to remember a key rule: the negative of a negative number is positive. This means if you see double negatives, like in e.g., - (-5), then the negatives 'cancel' out, making it a positive 5.
Understanding how to manipulate negative numbers is crucial for solving many kinds of mathematical problems, especially when working with inequalities.

Number Comparison

Comparing numbers is a fundamental mathematical skill. When comparing any two numbers, you need to determine which one is larger, smaller, or if they are equal. This is crucial in density concepts like inequalities.
There are a few key points to remember when comparing numbers:

• Positive numbers are always greater than negative numbers.
• Among solely positive numbers or solely negative numbers, the one with the higher absolute value is greater.

Using the numbers in the example: we need to compare 5 and -10. Since 5 is a positive number and -10 is a negative number, 5 is always greater than -10. This concept allows us to insert the correct inequality symbol, making sure our statements reflect realistic relationships between values. With practice, this process becomes almost instinctive.

Simplifying Expressions

Simplifying expressions is an essential skill that helps make complex mathematical problems more manageable. Simplifying means reducing an expression into its simplest form by performing all possible operations and combining like terms.
For expressions with negative numbers or nested operations, this can mean:

• Removing any double negatives, since -(-x) simplifies to x.
• Combining like terms with consistent sign rules.

In our example, the expression -(-5) simplifies directly to 5, because applying the negative of a negative turns it into a positive.
This simplification is a critical step for setting up and solving inequalities properly, as it ensures you have the correct numbers to compare. Mastering these simplification techniques will strengthen your overall ability to work efficiently with algebraic expressions.