Problem 54 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-1 \frac{1}{3} \text { and }-1.75$$

Step-by-Step Solution

Answer

-1.33 or -1 1/3 is greater than -1.75, the two inequalities are -1.33 > -1.75 and -1.75 < -1.33

Step 1: Convert the Mixed Fraction to Decimal

Convert \(-1 \frac{1}{3}\) into decimal form. This can be done by diving 1 by 3 to get \(0.333\) and then subtracting this from -1 to get \(-1.33\).

Step 2: Plot on the Number Line

Plot \(-1.33\) and \(-1.75\) on a number line. \(-1.75\) will be less than \(-1.33\) on the number line since it's further to the left.

Step 3: Write Inequalities

Now it's time to write the two inequalities, based on the positions of both numbers on the number line: \(-1.33 > -1.75\) and \(-1.75 < -1.33\)

Key Concepts

Understanding Inequalities

Inequalities help us compare two numbers, showing whether one number is greater than, lesser than, or equal to another number. The two primary symbols used are the greater than sign ">" and the less than sign "<". For example, if we say

• \(-1.33 > -1.75\)
• \(-1.75 < -1.33\),

we mean that \(-1.33\) is greater than \(-1.75\), while \(-1.75\) is less than \(-1.33\). You can think of it as an arrow pointing to the smaller number.
Visualizing these inequalities can be made easier when you plot the numbers on a number line, where numbers to the left are smaller.
This method gives a clear picture of the relative size of numbers! When working through math problems, always check your inequalities on a number line to verify your answers.

What Are Mixed Fractions?

A mixed fraction, also known as a mixed number, is a combination of a whole number and a fraction. For example, \(-1 \frac{1}{3}\) represents a whole number \(-1\), combined with a fractional part, \(\frac{1}{3}\).
To work with mixed numbers in calculations, such as comparing them on a number line, it's often easier to convert them into improper fractions or decimals.

• To convert \(-1\frac{1}{3}\) into a decimal, we can divide the fraction \(\frac{1}{3}\) which equals approximately \(0.333\), and then subtract that from \(-1\), leading to \(-1.33\).
• This process simplifies calculations and makes it easier to visualize on a number line.

By understanding the concept of mixed numbers, you make it easier for yourself to perform arithmetic operations and draw comparisons.

Decimal Conversion

Decimal conversion is transforming numbers into their decimal form.
This process is particularly useful when you need to perform operations that involve comparing numbers or finding their positions on a number line.
Here, it is crucial to understand how fractions convert into decimals:

• For instance, the fraction \(\frac{1}{3}\) is equal to the decimal \(0.333\) because when you divide 1 by 3, you get this repeating decimal.
• When combined with a whole number in mixed fractions, it is subtracted from that whole number. For example, \(-1\) minus \(0.333\) gives \(-1.33\).

Converting to decimal form is a powerful tool in simplifying math problems, making it easier to apply concepts like inequalities and place numbers on a number line accurately.
Use calculators for complex calculations and practice regularly to brush up on decimal skills.