Problem 7 - Algebra and Trigonometry

Question

7-8 Place the correct symbol \((<,>, \text { or }=)\) in the space. $$ \begin{array}{lllll}{\text { (a) } 3} & {\frac{7}{2}} & {\text { (b) }-3} & {-\frac{7}{2}} & {\text { (c) } 3.5} & {\frac{7}{2}}\end{array} $$

Step-by-Step Solution

Answer

(a) <, (b) >, (c) =

Step 1: Compare Values for 3 and \(\frac{7}{2}\)

First, convert the fraction to a decimal or compare it with an integer directly. The fraction \(\frac{7}{2}\) is equal to 3.5. Since 3 is less than 3.5, we place the symbol \(<\) between them. Thus, 3 < \(\frac{7}{2}\).

Step 2: Compare Values for -3 and \(-\frac{7}{2}\)

Next, compare the values of -3 and \(-\frac{7}{2}\). Again, convert \(-\frac{7}{2}\) to its decimal equivalent, which is -3.5. Since -3 is greater than -3.5, we place the symbol \(>\) between them. Therefore, -3 > \(-\frac{7}{2}\).

Step 3: Compare Values for 3.5 and \(\frac{7}{2}\)

Finally, compare 3.5 with the fraction \(\frac{7}{2}\). Since \(\frac{7}{2}\) equals 3.5, these two values are equal. We place the symbol \(=\) between them. Thus, 3.5 = \(\frac{7}{2}\).

Key Concepts

Inequalities

Inequalities are mathematical statements used to compare two values or expressions. They point out whether one value is less than, greater than, or equal to another. We use symbols to express these relationships:

• \(<\) for "less than"
• \(>\) for "greater than"
• \(=\) for "equal to"

Inequalities come in handy when dealing with a variety of mathematical problems, providing a way to express restrictions, limitations, or comparisons between values. In our exercise, we use inequalities to determine the relations between given numbers, involving both rational and decimal forms. By comparing each pair of numbers step by step, we can logically deduce which inequality symbol fits between them.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). Not only do they encompass fractions, but they also include integers (since any integer \(x\) can be written as \(\frac{x}{1}\)).
For example, in our exercise, the fraction \(\frac{7}{2}\) is a rational number. Rational numbers are hugely important in mathematics, as they allow for precise representations and calculations involving division. While working with rational numbers, converting between fractions and decimals is a common task, helping to ease comparison and operations.

Decimals

Decimals are another way to express numbers, using a point to separate the whole number from the fractional part. They are beneficial because they can represent fractions in a more visually intuitive way for many people, particularly when fractions have denominators like 10, 100, 1000, etc.
In the original exercise, converting fractions like \(\frac{7}{2}\) to a decimal form \(3.5\) simplifies the comparison process. Decimals help highlight the size relationships between numbers. When presented as decimals, it's often easier to directly compare number values to see which is greater or less, a crucial step for solving inequalities.

Fraction Conversion

Fraction conversion is the process of changing a fraction to a decimal or vice versa. This skill is crucial for comparing numbers and solving mathematical problems efficiently.

• To convert a fraction to a decimal, divide the numerator by the denominator.
• For example, \(\frac{7}{2}\) converts to \(3.5\) by dividing 7 by 2.
• Conversely, for decimals, find a fraction representation by using place value, such as \(0.75\) becoming \(\frac{75}{100}\).

Mastering fraction conversion is essential for many mathematical operations. It allows for easier calculations and comparisons, as different contexts might require different formats for clarity and simplicity. Knowing how to switch between forms effortlessly helps tackle a variety of problems efficiently.