Problem 25 - Algebra 1: Concepts and Skills

Question

Graph the numbers on a number line. Then write two inequalities that compare the numbers. \(-1,-6\)

Step-by-Step Solution

Answer

Placed on a number line, it is clear that \(-1 > -6\) and \(-6 < -1\).

Step 1: Plot the Numbers on a Number Line

On a number line, locate the spot corresponding to -1 and mark it. Next, locate the position for -6 and mark it. Remember that numbers to the right are greater while those to the left are lesser.

Step 2: Write Inequality Comparisons

Write two inequalities comparing the numbers. Taking a look at the number line, -1 is clearly to the right of -6, which means it is greater. Hence, the inequalities will be \(-1 > -6\) and \(-6 < -1\).

Key Concepts

Inequalities

Inequalities are mathematical expressions used to compare two values. They tell us whether one number is larger, smaller, or possibly the same as another number. When using inequalities, we utilize symbols such as:

• ">" means greater than.
• "<" means less than.
• ">=" means greater than or equal to.
• "<=" means less than or equal to.

The exercise given asked us to compare the numbers i.e., -1 and -6 using inequalities. By seeing where they fall on a number line, it was clear to say that -1 > -6 and -6 < -1. This means -1 is greater than -6 because, in any set of numbers, the number that appears on the right is always greater if it's farther from zero. This exercise highlights that inequalities are a critical tool in understanding how numbers relate to each other.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign "-". They are crucial in mathematics as they help describe values below a defined reference point, such as zero on the number line. For instance:

• A negative number like -1 indicates the position one unit below zero.
• -6 indicates a position six units below zero and hence is less than -1.

When dealing with negative numbers, the number with greater absolute value is further left on the number line and thus is less. For example, between -1 and -6, even though 6 is larger as a positive number, -6 is less than -1 in negative terms. Understanding how negative numbers are ordered is essential when solving inequalities since the rules differ from positive numbers.

Number Line Concepts

The number line is a visual representation to demonstrate the relationship between numbers. It's a straight line where each point corresponds to a number.
This tool is invaluable for illustrating numeric concepts, especially when dealing with inequalities and negative numbers.

• The number line extends indefinitely in both directions, with numbers increasing to the right and decreasing to the left.
• By plotting -1 and -6 on a number line, we can visually discern that -1 is to the right of -6.

This rightward placement indicates that -1 is greater than -6.
When graphing, the distance a number is from zero helps us understand its relative size in positive terms even as it dips into the negatives, simply remember, the smaller the absolute value closer it is to zero.
Grasping this concept is vital for working with negative values and inequalities.