Problem 63 - Introductory Algebra for College Students

Question

Determine whether each inequality is true or false. $$-5 \geq-13$$

Step-by-Step Solution

Answer

The inequality is True. \(-5\) is greater than \(-13\).

Step 1: Understand the inequality

An inequality compares two values, generally showing that one value is lesser or greater than the other. Here we need to understand the 'greater than or equal to' (\(\geq\)) sign. It denotes that the quantity before it is either greater than or equal to the quantity after it. \newline In \(-5 \geq -13\), it means \(-5\) should either be greater or equal to \(-13\).

Step 2: Compare the Numbers

It's crucial to understand that on a number line, a number is greater if it is to the right. So, although \(-5\) is numerically 'smaller' than \(-13\), it is actually greater because it lies more to the right on the number line. \newline As such, \(-5\) is indeed greater than \(-13\).

Key Concepts

Number Line

A number line is a visual representation that helps us understand the order and magnitude of numbers. It can be a straightforward tool for comparing numbers and solving inequality problems. Imagine the number line as a horizontal line with numbers evenly spaced along it. The larger numbers are always located to the right, while smaller numbers are to the left.

When you look at a number line, each number has a fixed position based on its value. For example, moving from left to right increases the value we are observing. This is why negative numbers like \(-13\) appear to the left of numbers like \(-5\). Even though \(-5\) appears smaller in magnitude, it is actually greater because it's located to the right of \(-13\) on the line.

So whenever you're in doubt about the relation between two numbers, using a number line can be a simple and effective way to visualize their relationship.

Greater Than or Equal To

The inequality sign \(\geq\) stands for "greater than or equal to." It is used in mathematics to compare two values. The left-hand side of this symbol must be either greater than or equal to the right-hand side to satisfy the inequality. This means that any number on the left can be equal to or further to the right on a number line compared to the number on the right.

Let's take the example of \(-5 \geq -13\). Here, we're trying to determine if \(-5\) is greater than or equal to \(-13\).

If you place these numbers on a number line, \(-5\) is indeed to the right of \(-13\). This shows \(-5\) is greater, thus satisfying the 'greater than' part of the inequality. Understanding this visual can help resolve doubts when determining if an inequality involving this symbol is true or false.

Comparing Numbers

Comparing numbers is a fundamental math skill. It involves determining which number is larger, smaller, or if both numbers are equal. This process is essential in working with inequalities like \(-5 \geq -13\).

Here's how to do it effectively:

• First, visualize or draw a number line if you're unsure. This helps in setting a clear picture of where each number belongs.
• Check the numbers to see which lies further to the right. Numbers to the right are always greater.
• Remember that with inequalities (\(\geq\), \(\leq\), etc.), you're not just looking for greater value but also considering the context of 'greater than or equal to.'

Comparing negative numbers can sometimes be tricky because they decrease in value the more you move to the left. In the case of \(-5\) and \(-13\), even though \(-5\) might look numerically smaller, it is actually larger than \(-13\) due to its position on the number line.