Problem 69 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ -\frac{11}{15} \quad-0.73 $$

Step-by-Step Solution

Answer

Insert \(-\frac{11}{15} < -0.73\) to make the statement true.

Step 1: Understand the Problem

The problem is asking us to compare two numbers: \(-\frac{11}{15}\) and \(-0.73\). We need to determine if \(-\frac{11}{15} < -0.73\) or \(-\frac{11}{15} > -0.73\).

Step 2: Convert the Fraction to Decimal

Convert \(-\frac{11}{15}\) to a decimal. Divide 11 by 15 to get approximately 0.7333. Thus, \(-\frac{11}{15} \approx -0.7333\).

Step 3: Compare the Decimals

Compare the decimals \(-0.7333\) and \(-0.73\). Since \(-0.7333\) is slightly less than \(-0.73\) (as more negative values are smaller), \(-0.7333 < -0.73\).

Step 4: Insert the Correct Symbol

Since \(-\frac{11}{15} < -0.73\), the correct symbol to insert is \(<\).

Key Concepts

Decimal Conversion

When comparing numbers, especially fractions and decimals, the first step often involves converting the fractions to decimal form. This allows for straightforward comparison using decimal arithmetic that most are familiar with.
To convert a fraction to a decimal, divide the numerator by the denominator. For instance, if we have \(\frac{11}{15}\), we perform the division 11 divided by 15. This gives us approximately 0.7333. Since the fraction \(-\frac{11}{15}\) is negative, the equivalent decimal is also negative, thus it's \(-0.7333\).

• Step-by-step conversion involves simple division.
• The result is a repeating or terminating decimal.
• It’s important to keep track of the negative signs.

This process is crucial in many math problems, as it provides a common ground for comparing numbers efficiently.

Negative Numbers

Negative numbers can be tricky, but they follow specific rules that, once understood, make them much easier to work with. Negative numbers are less than zero, and the further away they are from zero, the smaller their value.
In our comparison problem, both numbers are negative: \(-\frac{11}{15}\) and \(-0.73\).

• Negative values increase to zero as they decrease numerically.
• Think of it as debt; more negative means more debt.
• Larger negative numbers are smaller in value than smaller negative numbers.

This is why \(-0.7333\) is actually less than \(-0.73\), because \(-0.7333\) is further from zero.

Comparing Fractions and Decimals

After converting fractions to decimals, comparison becomes more straightforward. In essence, you now have two decimal numbers that you can compare just by looking at which is larger or smaller along the number line.
In this specific problem, comparing \(-0.7333\) with \(-0.73\) shows that \(-0.7333\) is indeed less than \(-0.73\).

• When two decimals start with the same numbers, continue checking each subsequent decimal place.
• Negative decimals nearer to zero are larger than those further away.
• Visualize these on a number line for clarity.

This comparative method is powerful and simple when dealing with math inequalities, helping gauge which numbers are larger or smaller efficiently, especially as part of students' homework with textbook exercises.