Problem 37 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ \frac{2-x}{\sqrt{9-6 x}}>0 $$

Step-by-Step Solution

Answer

The solution is \(-\infty < x < \frac{3}{2}\).

Step 1: Identify the Domain of the Inequality

The inequality involves the square root function in the denominator, so we need to ensure that the expression under the square root, \(9 - 6x\), is greater than zero. Solve this inequality: \(9 - 6x > 0\).

Step 2: Solve the Domain Inequality

Rearrange the inequality \(9 - 6x > 0\) to find \(-6x > -9\). Divide both sides by -6, remembering to flip the inequality sign: \(x < \frac{3}{2}\). Therefore, the domain is \(-\infty < x < \frac{3}{2}\).

Step 3: Analyze the Numerator and Denominator

We need the fraction \(\frac{2-x}{\sqrt{9-6x}}\) to be greater than zero, meaning the numerator and the denominator must have the same sign for the expression to be positive. Analyze \(2-x > 0\), assuming \(\sqrt{9-6x} > 0\) from the domain condition. Solve \(2-x > 0\) to get \(x < 2\).

Step 4: Determine Overlapping Interval

From Steps 2 and 3, we know that \(x < \frac{3}{2}\) and \(x < 2\). The most restrictive condition is \(x < \frac{3}{2}\). Therefore, the solution to the inequality is \(-\infty < x < \frac{3}{2}\).

Step 5: Conclude the Solution

The inequality is satisfied for all \(x\) such that \(x\) is less than \(\frac{3}{2}\), as this ensures both the numerator \(2-x\) and the square root denominator remain positive, and the fraction itself remains greater than zero.

Key Concepts

Understanding the Inequality Domain

When tackling inequalities like \(\frac{2-x}{\sqrt{9-6x}} > 0\), understanding the domain is crucial. The domain is all the possible values that \(x\) can take without making any part of the inequality undefined. In this exercise, the square root in the denominator, \(\sqrt{9-6x}\), imposes a restriction. Square roots are only defined for non-negative numbers, so we set the expression under the square root greater than zero: \(9 - 6x > 0\). Solving this gives \(x < \frac{3}{2}\). Hence, \(x\) can take any real number value less than \(\frac{3}{2}\), providing our domain: \(-\infty, \frac{3}{2}\). This ensures our denominator never becomes zero or negative, keeping the expression valid.

Numerator and Denominator Analysis

In an inequality of the form \(\frac{2-x}{\sqrt{9-6x}} > 0\), ensuring the numerator and the denominator have the same sign is essential. This guarantees the expression remains positive. Let's break it down:

• Numerator: \(2-x\): We require \(2-x > 0\) which means \(x < 2\).
• Denominator: \(\sqrt{9-6x}\): From domain analysis, we only consider \(x\) values such that \(\sqrt{9-6x} > 0\).

Both parts need to satisfy these conditions to ensure that \(\frac{2-x}{\sqrt{9-6x}}\) is positive, leading us to analyze potential overlaps between these conditions.

Finding the Overlapping Interval

Determining the overlapping interval involves finding values of \(x\) that satisfy both conditions expressed in previous steps. From the domain, we obtained the restriction \(x < \frac{3}{2}\) to ensure the square root remains positive.Additionally, from analyzing the numerator, we concluded \(x < 2\). The overlapping interval between these conditions is restricted by the most limiting condition, which is \(x < \frac{3}{2}\). This interval provides the set of \(x\) values needed to keep the full expression positive. This overlap shows us where both the numerator and the denominator are positive, ensuring the fraction itself is greater than zero.

Summarizing the Inequality Solution Set

The solution set of the inequality is composed of all \(x\) values that satisfy the derived conditions. Based on the overlapping interval identified, the solution set is in the range \(-\infty, \frac{3}{2}\). This means any real number less than \(\frac{3}{2}\) will satisfy the inequality. For better understanding:

• The domain requirement ensures no division by zero or undefined square roots.
• Both numerator and denominator must maintain positive signs to fulfill the inequality \(\frac{2-x}{\sqrt{9-6x}} > 0\).
• The intersection of conditions results in the final solution set: \(x \in (-\infty, \frac{3}{2})\).

This concise solution set reflects all the allowable \(x\) values making the expression valid and above zero.