Problem 54 - Algebra 2

Question

If \(5 < a < 7 < b < 14\) then which of the following best describes \(\frac{a}{b} ?\) A \(\frac{5}{7} < \frac{a}{b} < \frac{1}{2}\) B \(\frac{5}{14} < \frac{a}{b} < \frac{1}{2}\) C \(\frac{5}{7} < \frac{a}{b} < 1\) D \(\frac{5}{14} < \frac{a}{b} < 1\)

Step-by-Step Solution

Answer

The correct answer is D: \(\frac{5}{14} < \frac{a}{b} < 1\).

Step 1: Understand the Range of 'a' and 'b'

We are given that \(5 < a < 7\) and \(7 < b < 14\). This means 'a' is any number greater than 5 and less than 7, and 'b' is any number greater than 7 and less than 14.

Step 2: Determine the Possible Range for \(\frac{a}{b}\)

The smallest value of \(\frac{a}{b}\) occurs when 'a' is smallest and 'b' is largest. Thus, it could be \(\frac{5}{14}\). The largest value of \(\frac{a}{b}\) occurs when 'a' is largest and 'b' is smallest, which could be \(\frac{7}{7} = 1\).

Step 3: Compare with Answer Choices

Since the range for \(\frac{a}{b}\) is \(\frac{5}{14} < \frac{a}{b} < 1\) from Step 2, compare this range with the answer choices: A: \(\frac{5}{7} < \frac{a}{b} < \frac{1}{2}\)B: \(\frac{5}{14} < \frac{a}{b} < \frac{1}{2}\)C: \(\frac{5}{7} < \frac{a}{b} < 1\)D: \(\frac{5}{14} < \frac{a}{b} < 1\).

Step 4: Choose the Correct Range

The correct range that fits \(\frac{5}{14} < \frac{a}{b} < 1\) is answer choice D.

Key Concepts

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator, the denominator, or both. In this context, the expression \(\frac{a}{b}\) is considered a rational expression because both 'a' and 'b' represent some value or polynomial, even though they are simple variables in this instance. Understanding rational expressions involves recognizing the rules of arithmetic applied to fractions, which similarly apply to more complex rational expressions:

• The simplification of these expressions follows the same principles as simplifying basic fractions.
• We handle divisions and multiplications as usual but must always be wary of undefined expressions when the denominator equals zero.

In the exercise, \(\frac{a}{b}\) is bounded by constraints that turn it into just a value within a specified range, significantly simplifying the manipulation compared to more complex rational expressions in algebra.

Number Ranges

When dealing with inequalities, number ranges help define a set of possible values a variable can take on. This is crucial in problems involving rational expressions, like \(\frac{a}{b}\), since the range from which we select 'a' and 'b' affects the bounds of the expression itself.In the given problem, we had two key ranges:

• \(5 < a < 7\)
• \(7 < b < 14\)

These ranges provide critical limits within which these variables can fluctuate. The expression \(\frac{a}{b}\) can vary greatly depending on which values are chosen for 'a' and 'b' within their respective ranges. Identifying the extreme values within these ranges (both upper and lower bounds) allows us to determine the possible spread of the rational expression. By ensuring values never cross these boundaries, we maintain the integrity of the inequalities, allowing us to precisely measure and compare such expressions.

Problem Solving

Problem solving in the context of inequalities with rational expressions is about systematically narrowing down possible values. It often involves evaluating extreme cases or boundaries to make sound deductions about the expression in question.In this exercise, the task required analyzing extremes:

• To find the smallest value of \(\frac{a}{b}\), assume 'a' is at its minimum and 'b' at its maximum yielding \(\frac{5}{14}\).
• For the largest value, assume 'a' at its maximum and 'b' at its minimum, giving \(\frac{7}{7} = 1\).

By adopting these strategies, we ensure the derived range is tight and accurate. Compare the deduced range to multiple-choice offerings, choosing the one that matches perfectly. Problem solving here intertwines numerical analysis with logical deduction, aiming to find a precise answer through methodical reasoning.