- The absolute value \(|x|\) represents the magnitude of debt, so \(|-300| = 300\) and \(|-100| = 100\).
- Numerically, \(-300 < -100\), but the magnitude of the \$300 debt is greater than the \$100 debt.
- A \$100 debt is therefore a smaller or "lesser" debt than a \$300 debt.
- The correct term to fill the second blank is \(\boxed{\text{lesser}}\).
Explanation
- Analyzing the Inequality of Magnitudes
The inequality \(|-300| > |-100|\) compares the absolute values of two account balances. The absolute value \(|x|\) of a number \(x\) represents its distance from zero on the number line, or its magnitude. In financial terms, if a balance represents a debt, the negative sign indicates an amount owed, and the absolute value represents the actual size of that debt. Thus, \(|-300| = 300\) and \(|-100| = 100\). Since \(300 > 100\), we see that the magnitude of a \$300 debt is greater than the magnitude of a \$100 debt.
- Completing the Comparison Statement
Numerically, on the number line, negative numbers further from zero are smaller. Therefore, \(-300\) is less than \(-100\). However, in terms of debt, the "size" or magnitude of the debt is what matters. Since \(| -300 | = 300\) and \(| -100 | = 100\), a \$300 debt is a larger debt than a \$100 debt. Consequently, a \$100 debt is a smaller or "lesser" debt than a \$300 debt. The sentence structure requires us to clarify the numerical relationship versus the actual debt burden. Following standard mathematical terminology for comparing these debt amounts, the first blank refers to the numerical relationship (\(-300\) is mathematically less than \(-100\)) and the second blank uses the comparative "lesser" to describe the debt size.
- Conclusion
The inequality \(|-300| > |-100|\) demonstrates that while \(-300\) is numerically smaller than \(-100\), the magnitude of the debt represented by \(-300\) is greater than the magnitude of the debt represented by \(-100\). Therefore, the correct completion of the sentence is: 'Even though \(-300\) is numerically less than \(-100\), a \$100 debt is a lesser debt than a \$300 debt.' Given the options provided, the most appropriate term for the debt comparison in the second blank is 'lesser'. The final answer is \(\boxed{\text{lesser}}\).
Examples
Understanding absolute values is essential in personal finance. For instance, if you have two credit card balances, one at -\$500 and one at -\$200, comparing the absolute values \(|-500| = 500\) and \(|-200| = 200\) helps you identify which debt is larger (\(500 > 200\)). This helps in prioritizing debt repayment strategies, where you might choose to pay off the larger debt magnitude first to reduce total interest costs. Recognizing the difference between numerical value and magnitude is a fundamental skill for financial literacy.