- Subtract the right side to get \(\frac{4x-3}{x-2} \leq 0\).
- Identify critical points \(x = 3/4\) and \(x = 2\).
- Test the intervals \((-\infty, 3/4]\), \([3/4, 2)\), and \((2, \infty)\) to verify where the expression is non-positive.
- Include the point \(x = 3/4\) (as it makes the expression 0) and exclude \(x = 2\) (as it makes the denominator 0) to get the final solution \(\boxed{[3/4, 2)}\).
Explanation
- Analyzing the Inequality
The goal is to solve the rational inequality \(\frac{4x}{x-2} \leq \frac{3}{x-2}\). First, we note the domain constraint for the variable \(x\): the denominator cannot be zero, so \(x-2
eq 0\), which implies \(x
eq 2\). We proceed by isolating all terms on one side of the inequality to compare the expression with zero. Subtracting \(\frac{3}{x-2}\) from both sides gives: \[\frac{4x}{x-2} - \frac{3}{x-2} \leq 0\] Since the denominators are the same, we can combine the numerators: \[\frac{4x - 3}{x - 2} \leq 0\]
- Identifying Critical Points
To determine the intervals where the expression \(\frac{4x-3}{x-2}\) satisfies the inequality, we identify the critical points where the numerator is zero or the denominator is zero. Setting the numerator to zero: \(4x - 3 = 0 \Rightarrow x = 3/4\). Setting the denominator to zero: \(x - 2 = 0 \Rightarrow x = 2\). These points partition the real number line into three intervals: \((-\infty, 3/4]\), \([3/4, 2)\), and \((2, \infty)\).
- Testing Intervals and Final Solution
We now test each interval to check the sign of the rational expression \(\frac{4x-3}{x-2}\):
- For \(x < 3/4\) (e.g., \(x = 0\)): \(\frac{4(0) - 3}{0 - 2} = \frac{-3}{-2} = 1.5 > 0\). This interval does not satisfy the inequality.
- For \(3/4 < x < 2\) (e.g., \(x = 1\)): \(\frac{4(1) - 3}{1 - 2} = \frac{1}{-1} = -1 \leq 0\). This interval satisfies the inequality.
- For \(x > 2\) (e.g., \(x = 3\)): \(\frac{4(3) - 3}{3 - 2} = \frac{9}{1} = 9 > 0\). This interval does not satisfy the inequality.
Next, we check the endpoints: at \(x = 3/4\), the expression equals zero, which is allowed by the \(\leq\) inequality. At \(x = 2\), the expression is undefined, so this point must be excluded. Thus, the solution set is \([3/4, 2)\).
Examples
Consider a scenario where you are calculating the efficiency of a supply chain process. If the cost efficiency, represented by \(\frac{4x}{x-2}\), must remain lower than or equal to a baseline of \(\frac{3}{x-2}\), then you need to find the range of operation values (\(x\)) for which this is true. By solving this rational inequality, you determine that the efficiency remains within the acceptable range when \(x\) is between 0.75 and 2. Knowing these operational boundaries is essential for optimizing system performance and maintaining cost-effective operations.