Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. \(-2\) and \(-1\)

Short answer

The two numbers are graphed on the number line with \(-2\) to the left of \(-1\). The two inequalities comparing these numbers are \(-2 < -1\) and \(-1 > -2\).

Step-by-step solution

Understanding the number line

A number line is a visual representation where numbers are arranged in ascending order from left to right, with zero in the middle. Numbers to the right of zero are positive, and to the left are negative.

Graphing the numbers

Locate the numbers \(-2\) and \(-1\) on the number line. Plot a point at \(-2\) and another at \(-1\).

Writing the inequalities

An inequality is a statement that compares two values. Looking at the number line, \(-2\) is on the left of \(-1\), meaning \(-2\) is lesser than \(-1\). The two inequalities will then be \(-2 < -1\) and \(-1 > -2\).

Key concepts

Number Line

The number line is an essential tool for understanding mathematical concepts related to numbers and their positions relative to one another. It is a straight, horizontal line that has points which correspond to numbers. The center of the line is usually marked with zero, known as the origin, and it helps us differentiate between positive and negative numbers. To the right of zero, the numbers increase in value, representing positive numbers, while to the left, the numbers decrease, representing negative numbers.

To graph numbers on a number line, we start by identifying the location of each number. If we take the numbers (-2) and (-1), we would find the point marked (-2) and place a dot or a line above it to indicate the number's position. We then do the same for (-1). By placing these two numbers on the number line, we can visually understand which is greater or lesser.

Inequalities

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, equal to, or not equal to the other. They are crucial for expressing a range of possible solutions or for stating that one quantity is bounded by another. The inequality symbols include <, >, \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to).

When graphing the numbers (-2) and (-1) on a number line, we determine the inequalities by their positions. Since (-2) is to the left of (-1), it signifies that (-2) is less than (-1), which can be written as (-2 < -1). Conversely, we can also express that (-1) is greater than (-2), using the inequality (-1 > -2). These expressions provide a concise way to compare and describe the relationship between numbers.

Visual Representation of Numbers

A visual representation of numbers, such as using a number line, helps to make abstract concepts more concrete. It supports students in understanding the relative positions and magnitude of numbers. This visual tool is particularly effective when learning about positive and negative values, sequences, and the concept of absolute value.

For instance, when we plot the numbers (-2) and (-1) on a number line, we can see the actual distance between them and recognize that (-1) is one unit closer to zero than (-2). Such visual clues help in understanding not only the basic concept of ordering but also more complex operations like addition and subtraction on the number line.

Comparing Numbers

Comparing numbers is a fundamental mathematical skill that involves determining the relative size of two numbers. It is critical in various contexts, from everyday decision-making to solving complex equations. Using a number line is an effective method for comparing because it provides a clear, visual way to see which of two numbers is larger, smaller, or if they are equal, based on their positioning.

The comparison of the numbers (-2) and (-1) quickly reveals that (-1) is greater than (-2), since (-1) is positioned to the right of (-2) on the number line. Conversely, we say that (-2) is less than (-1) because it is on the left. This concept extends beyond just whole numbers and is applicable to fractions, decimals, and even irrational numbers.