Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. \(-7\) and 3

Short answer

The two inequalities are \( -7 < 3 \) and \( 3 > -7 \).

Step-by-step solution

Drawing the Number Line

Start by drawing a number line. The number line should include all integers between -7 and 3. Draw a point on the number line for each of the numbers, -7 and 3.

Positioning the Numbers

Since the numbers -7 and 3 are given, they should be marked appropriately on the number line. -7 is placed on the left side of the number line since it's smaller, and 3 is placed on the right side of the number line since it's bigger.

Writing Inequalities

The two numbers can now be compared using inequalities. From the number line, one can see that -7 is less than 3 and 3 is greater than -7. Thus, the two inequalities are \( -7 < 3 \) and \( 3 > -7 \).

Key concepts

Inequalities

Understanding inequalities is fundamental in algebra and representing real-world situations. Inequalities show how numbers relate to each other, indicating that one number is either less than, greater than, less than or equal to, or greater than or equal to another number.

When graphing numbers like (-7) and 3, we observe their relative positions and can determine that -7 is indeed less than 3. This is represented by the inequality (-7 < 3). Similarly, it’s correct to say that 3 is greater than -7, expressed as (3 > -7). These mathematical statements provide a concise way to compare values and are particularly useful when solving problems that involve ranges or limits, such as determining the allowable weights for parcels or calculating the boundaries of a solution set in algebra.

Number Line

A number line is an essential visual aid in mathematics that helps with the understanding of numbers and their relationships. The line consists of points that correspond to numbers, with the distance between points equaling to the difference in value between those numbers.

When graphing integers, such as -7 and 3, on a number line, we visually place them according to their value with negative integers to the left of zero, and positive integers to the right. This makes it clear to see that -7 lies to the left of 3, reinforcing our understanding of their relative sizes without involving complex calculations. Graphing numbers on a number line provides a straightforward method for comparing integers and learning the concept of absolute value.

Comparing Integers

Comparing integers is one of the most straightforward concepts in mathematics once you get the hang of it. It’s simply a matter of looking at two numbers and deciding which one is larger or smaller.

To effectively compare integers, remember that every positive number is greater than every negative number. For example, 3 is greater than -7, not just because 3 is positive and -7 is negative, but also because on the number line, 3 is positioned to the right of -7. Also, amongst positive integers, the one with the higher value is the larger one, whereas among negative integers, the one with the smaller absolute value (-1 compared to -3, for example) is larger. The visual aid of a number line is particularly helpful in comparing integers as it presents a clear and tangible way of determining which integer is larger or smaller.

Algebraic Concepts

Algebraic concepts are the building blocks for understanding more advanced math topics and for solving real-world problems. They include understanding variables, expressions, equations, functions, and inequalities.

Graphing numbers and inequalities on a number line embodies these algebraic concepts by translating them into a visual format. The idea is to not just see the numbers and operations as abstract symbols, but as parts of a larger picture that can be manipulated and understood spatially. This spatial representation is particularly useful when you're dealing with inequalities, as it allows you to visually identify solutions to an equation or compare the relative sizes of different numbers. By infusing these fundamental concepts of algebra into visual formats, like the number line, new learners can develop a stronger, more intuitive understanding of mathematics.