Problem 50 - Elementary Algebra

Question

Fill in the blank with \(<,=,\) or \(\geqslant\). -40 _____ -41

Step-by-Step Solution

Answer

-40 \\geqslant -41

Step 1: Compare the Absolute Values

First, we focus on the absolute values of the numbers. The absolute value of -40 is 40, and the absolute value of -41 is 41. Since 40 < 41, the number with the smaller absolute value is closer to zero or lesser in magnitude.

Step 2: Determine the Position on the Number Line

Consider where these numbers fall on the number line. A larger negative number indicates a position further left on the number line. Since -41 is further left than -40, it means -40 is greater than -41.

Step 3: Fill in the Blank with the Correct Symbol

Given the previous steps, we can fill in the blank with the correct inequality symbol. Since -40 is greater than -41, the correct symbol to use is \(-40 \geqslant -41\).

Key Concepts

Absolute Value

Understanding absolute value is essential when dealing with inequalities, especially with negative numbers. The absolute value of a number is its distance from zero on the number line, without considering direction. It's always non-negative. Think of it as measuring how far a number is from zero, regardless of whether it's positive or negative. For example:

• The absolute value of \( -40 \) is \( 40 \).
• The absolute value of \( -41 \) is \( 41 \).

When comparing numbers, we often check which has a smaller absolute value to determine which is numerically closer to zero. This is useful because numbers closer to zero are larger in terms of their position on the number line, even if they are negative. Remember, lesser absolute value can mean greater numerical closeness to zero.

Number Line

A number line is a helpful visual tool for understanding the relative positions of numbers, especially when they involve negative values. It's a straight line where each point represents a number. Numbers increase as you move to the right and decrease as you move to the left.

• Zero sits at the center and acts as a pivot point.
• Negative numbers extend to the left of zero, showing smaller values than positive numbers.
• Positive numbers stretch to the right, indicating increasing values.

When you compare two negative numbers like \( -40 \) and \( -41 \), look at their positions on the number line. On it, \( -41 \) is further to the left than \( -40 \), indicating \( -41 \) is smaller in value. This visualization helps solidify the idea that greater negative numbers lie further left, while smaller (or less negative) numbers are closer to zero.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (\( - \)). They play a crucial role in mathematics, especially in inequalities.

• Negative numbers represent the opposite of positive numbers, often used to signify loss or decrease.
• When you compare negative numbers, the one with the smaller numerical value is actually greater because it's less negative.

For example, consider \( -40 \) and \( -41 \). Though \( 41 \) is numerically larger than \( 40 \), \( -41 \) is a smaller number because it's farther from zero on the negative side. Understanding this helps in solving inequality problems involving negative numbers. More negative means further left on the number line, and hence lesser in value compared to less negative numbers.