Question

Use \(<\) or \(>\) to write a true statement. $$-30 ?-16$$

Short answer

Answer: -30 < -16

Step-by-step solution

Comparing Magnitudes

Since both -30 and -16 are negative numbers, we can analyze which of the two numbers is farther from 0. That number, being more negative, will have a greater magnitude. In this case, since -30 is farther from 0 than -16, its magnitude is greater.

Choose the Correct Inequality Symbol

Since -30 has a greater magnitude (is farther from 0), it is less than -16. Therefore, we write the inequality as: $$-30 < -16$$

Key concepts

Comparing Negative Numbers

When comparing negative numbers, it's essential to understand that the number closer to zero is considered larger. This may seem counterintuitive because with positive numbers, the farther from zero, the larger the value. However, in the world of negative numbers, the rules are reversed. If you are given two negative numbers, such as \( -30 \) and \( -16 \), it is important to visualize these on a number line.

Imagine a horizontal line with zero in the middle. To the left are negative numbers, and to the right are positive numbers. Negative numbers increase in value as they move closer to zero. Hence, \( -16 \) is closer to zero and therefore larger than \( -30 \). This is why we say that \( -30 < -16 \). Here's an easy tip: think of negative numbers like a game where the goal is to get as close to zero as possible. The number that achieves this is the 'winner' or the greater number.

Magnitude of Numbers

Often, students find the term 'magnitude' confusing when it applies to both positive and negative numbers. The magnitude of a number is essentially a measure of how big a number is regardless of its sign. For positive numbers, it's straightforward—the bigger the number, the larger its magnitude. When dealing with negative numbers, remember that magnitude is still about how far a number is from zero, not whether it is more or less than another number.

This is why we describe \( -30 \) as having a greater magnitude than \( -16 \), because it is further away from zero on the number line. However, when it comes to the value of the number in terms of greater or less than, \( -16 \) is greater because it is less negative. A good exercise to get familiar with magnitudes is to practice plotting negative and positive numbers on a number line and seeing their relative distances from zero.

Inequality Problem Solving

Solving inequality problems can seem daunting, but with a clear understanding of number lines and magnitudes, it becomes much simpler. The first step in inequality problem solving involving negative numbers is to accurately compare the magnitudes, as you would with positive numbers. Remember, when the signs are the same (both positive or both negative), the number closer to zero is considered the smaller number.

In the case of \( -30 \) and \( -16 \) from our original problem, since both numbers are negative, we find that \( -16 \) is closer to zero. Thus, the correct inequality symbol to create a true statement is 'greater than' when considering their values, or \( -30 < -16 \). To reinforce these concepts, students may benefit from using a number line when comparing numbers and practicing with a variety of inequality problems to see the patterns and reinforce their understanding.