Problem 72 - Elementary and Intermediate Algebra

Question

Insert one of the symbols \(>,<,\) or \(=\) in the blank to make each statement true. $$ -5.5 \quad \text{_____}\quad-5 \frac{1}{2} $$

Step-by-Step Solution

Answer

The statement is true with \(=\).

Step 1: Rewrite Mixed Number

Rewrite the mixed number \(-5 \frac{1}{2}\) as an improper fraction. \(-5 \frac{1}{2} = -\left(5 + \frac{1}{2}\right) = -\frac{11}{2}\).

Step 2: Decimal Equivalent

Convert \(-\frac{11}{2}\) into a decimal to compare it easily with \(-5.5\). Divide 11 by 2 to get \(-5.5\).

Step 3: Compare the Numbers

Now that both numbers, \(-5.5\) and \(-5.5\), are in decimal form, observe that they are equal. Thus, insert \(=\) in the blank.

Key Concepts

Improper Fractions

An improper fraction is where the numerator is greater than or equal to the denominator. This means the fraction represents a whole number and a part of another whole number, or it might just represent a whole number itself if the numerator is exactly a multiple of the denominator.

• For example, in the fraction \(-\frac{11}{2}\), the numerator 11 is greater than the denominator 2, making it an improper fraction.
• To convert a mixed number into an improper fraction, you multiply the whole number by the denominator and add the numerator of the fraction part.

In the example, \(-5 \frac{1}{2}\) was converted by calculating: \(-5 \times 2 = -10\), then add 1 for \(-11\). Therefore, we have \(-\frac{11}{2}\) as the improper fraction.

Decimal Conversion

Decimal conversion is the process of changing a fraction into a decimal. This helps in comparing numbers more easily, as decimal numbers can be compared simply by looking at their value from left to right.

• Let's look at the improper fraction \(-\frac{11}{2}\). Converting this into a decimal involves dividing the numerator by the denominator: 11 divided by 2 results in 5.5; thus, the conversion gives us \(-5.5\).

Why convert? This allows straightforward comparisons because decimals have a consistent place value. So, comparing \(-5.5\) to \(-5.5\) becomes a simple observation of equality.

Mixed Numbers

Mixed numbers combine a whole number and a proper fraction. A proper fraction is where the numerator is less than the denominator. If you have \(-5 \frac{1}{2}\), it means 5 whole units and half of another unit.

• Mixed numbers are useful when you need to express a value that sits between two whole numbers.
• Going from a mixed number to an improper fraction may simplify calculations, especially when comparing or performing operations like addition or subtraction.

How does this work in our example? By converting the mixed number \(-5 \frac{1}{2}\) to the improper fraction \(-\frac{11}{2}\), we can further convert to a decimal form \(-5.5\). This conversion process clarifies comparisons, revealing that the original numbers are equal.