Problem 31 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Convert Mixed Number

In problem (a), convert \(\frac{7}{2}\) to a mixed number or decimal. Since \(\frac{7}{2} = 3.5\), we compare 3 to 3.5.

Step 2: Compare Values in (a)

Compare the values 3 and 3.5. We know 3 is less than 3.5, so the correct symbol is \(<\).

Step 3: Absolute Values for (b)

For problem (b), consider the positive values of \(-3\) and \(-\frac{7}{2}\). We need to calculate and compare \(-3\) and \(-3.5\).

Step 4: Compare Values in (b)

For \(-3\) and \(-3.5\), \(-3\) is greater than \(-3.5\), so we place \(>\).

Step 5: Convert to Decimal for (c)

In (c), compare 3.5 and \(\frac{7}{2}\), where \(\frac{7}{2} = 3.5\), the same as the decimal.

Step 6: Compare Values in (c)

Since 3.5 equals 3.5, the correct symbol is \(=\).

Key Concepts

Mixed Numbers

Mixed numbers consist of an integer and a proper fraction. They often appear when dealing with fractions greater than one. To understand mixed numbers, it's vital to learn how to convert between mixed numbers and improper fractions.
For instance, if you have a mixed number like 3 ½, you can convert it into an improper fraction by multiplying the whole number by the denominator and adding this to the numerator:

• Multiply the whole number (3) by the denominator (2): 3 × 2 = 6
• Add the result to the numerator (1): 6 + 1 = 7
• Place this result over the original denominator: 7/2

Thus, 3 ½ becomes 7/2. This method ensures simple comparisons or calculations because it's easier to work with improper fractions sometimes in mathematical equations.

Absolute Values

Absolute value refers to the non-negative value of a number, regardless of its sign. It measures the distance of a number from zero on a number line. For example, both 3 and -3 have an absolute value of 3.
When comparing negative numbers, it's beneficial to first consider their absolute values. This helps in assessing their size without the negative sign getting in the way. However, remember that the further left a negative number is on the number line, the smaller it is relative to other negative numbers.
In practice, to compare -3 and -3.5: Calculate their absolute values first.

• Absolute value of -3 is 3
• Absolute value of -3.5 is 3.5

By using absolute values, you can predict which number is larger, but in comparing negatives themselves, -3 is indeed greater than -3.5.

Decimal Conversion

Decimal conversion is the process of transforming fractions or mixed numbers into decimal form. This process makes it easier to perform mathematical operations and comparisons.
Take the example from our exercise: converting \(\frac{7}{2}\) into a decimal. To do this, divide the numerator by the denominator: \(7 \div 2 = 3.5\).
Once in decimal form, numbers are simple to compare—like comparing weights or measuring lengths in a standardized form. By converting every fraction to its decimal equivalent, you can uniformly evaluate their value against other numbers.

Inequality Symbols

Inequality symbols help us express relationships between numbers. The common symbols include:\

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

When comparing two numbers, like those in the exercise, inequality symbols provide a concise way to express which one is larger or smaller.
For example, given two numbers 3 and 3.5, you can use the less than symbol to indicate that 3 is smaller than 3.5. Therefore, 3 \( < \) 3.5.
Understanding and using these symbols correctly is crucial in math since they help determine the right relationship in equations and inequalities.