Problem 43 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$ -3 \text { and }-\frac{7}{2} $$

Step-by-Step Solution

Answer

The plotted numbers on the number line show that -3 is greater than -7/2. Therefore, the inequalities are -3 > -7/2 and -7/2 < -3.

Step 1: Draw a Number Line

Draw a horizontal line and label it as the number line. Mark the integer values on the number line clearly, ensuring that -3 and -7/2 (-3.5) will be included.

Step 2: Plot the Numbers

Plot the numbers on the number line. -3 and -7/2 should be marked.

Step 3: Compare the Numbers

Looking at the number line, it can be seen that -3 is to the right of -7/2, which means -3 is greater than -7/2 as negative numbers become larger as we move to the right on the number line.

Step 4: Write Inequalities

Since -3 is greater than -7/2 we can write the inequality as -3 > -7/2. We can also write the inequality in the opposite direction, -7/2 < -3.

Key Concepts

Comparing Integers

Understanding how to compare integers is essential when dealing with numbers on the number line. Integers are whole numbers that include negative numbers, zero, and positive numbers. When graphing integers, we follow the convention that numbers increase in value as we move to the right on the number line and decrease as we move to the left.

For example, let's compare -3 and -4. Since -3 is to the right of -4 on the number line, -3 is greater than -4. This is expressed as \( -3 > -4 \). Similarly, when we compare -3 and -2, we find that -3 is to the left of -2, which means -3 is less than -2, or \( -3 < -2 \).

An easy way to remember this is to think of the number line as a horizontal street: addresses increase as you move towards the positive direction, thus the 'right' side has larger numbers, and the 'left' side has the smaller numbers.

Plotting Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. When plotting rational numbers on a number line, we need to identify their precise location based on their value.

Take the rational number \( -\frac{7}{2} \) as an example. To plot this number, find the two whole integers it lies between—in this case, -3 and -4—and determine its exact position. Since \( -\frac{7}{2} \) equates to -3.5, we would place it halfway between -3 and -4 on the number line.

Comparison of rational numbers follows the same rule as integers: a number to the right is greater than one to its left. Thus, \( -\frac{7}{2} \) is less than -3 because it is located to the left of -3 on the number line. This understanding is crucial when mathematically representing situations in fields such as economics, statistics, and physics, where precision is key.

Writing Inequalities

Writing inequalities is a mathematical way of comparing two values to show that one value is greater than, less than, equal to, or not equal to another value. Inequalities are represented by signs such as '<' (less than), '>' (greater than), '\leq' (less than or equal to), and '\geq' (greater than or equal to).

Let's consider the numbers -3 and \( -\frac{7}{2} \). Based on their position on the number line, we can write two inequalities. Since -3 is to the right of \( -\frac{7}{2} \), we conclude that -3 is greater than \( -\frac{7}{2} \), and write the inequality as \( -3 > -\frac{7}{2} \). Conversely, we can also describe this relationship by saying \( -\frac{7}{2} \) is less than -3, written as \( -\frac{7}{2} < -3 \).

Inequalities are a foundational concept in not only algebra but also in various real-world scenarios such as determining budget limits, temperature ranges, and quantities in inventory management. This utility makes understanding and writing inequalities a vital skill for students.