- The given inequality \(\frac{4x}{x-2} \leq \frac{3}{x-2}\) is simplified by subtracting the right side, resulting in \(\frac{4x-3}{x-2} \leq 0\).
- Critical points are identified as \(x = 3/4\) and \(x = 2\) (where the expression is zero or undefined).
- Testing intervals reveals that the expression is negative only in the range \([3/4, 2)\).
- The solution set to the inequality is \([\frac{3}{4}, 2)\).
- Final Answer: \(\boxed{[\frac{3}{4}, 2)}\)
Explanation
- Problem Analysis and Domain Restriction
We are given the rational inequality \(\frac{4x}{x-2} \leq \frac{3}{x-2}\). Our goal is to find all real values of \(x\) that satisfy this inequality. A key constraint is that the expression is undefined when the denominator equals zero, so we must have \(x-2
eq 0\), which means \(x
eq 2\). We should avoid multiplying by \(x-2\) directly because its sign is unknown and multiplying by a negative number would reverse the inequality sign. Instead, we move all terms to one side to compare the expression to zero.
- Simplifying the Inequality and Identifying Critical Points
Subtracting \(\frac{3}{x-2}\) from both sides, we get:
\[\frac{4x}{x-2} - \frac{3}{x-2} \leq 0\]
Combining these into a single fraction:
\[\frac{4x - 3}{x - 2} \leq 0\]
Now, we find the critical points by setting the numerator and denominator to zero:
- Numerator: \(4x - 3 = 0 \Rightarrow x = \frac{3}{4}\)
- Denominator: \(x - 2 = 0 \Rightarrow x = 2\)
These critical points divide the real number line into three intervals: \((-\infty, \frac{3}{4}]\), \([\frac{3}{4}, 2)\), and \((2, \infty)\).
- Testing Intervals and Determining the Solution Set
To determine which intervals satisfy the inequality, we test a value from each interval in the expression \(f(x) = \frac{4x - 3}{x - 2}\):
- For \(x \in (-\infty, \frac{3}{4})\), test \(x = 0\):
\[f(0) = \frac{4(0) - 3}{0 - 2} = \frac{-3}{-2} = 1.5\]
Since \(1.5 > 0\), this interval does not satisfy the inequality.
- For \(x \in (\frac{3}{4}, 2)\), test \(x = 1\):
\[f(1) = \frac{4(1) - 3}{1 - 2} = \frac{1}{-1} = -1\]
Since \(-1 \leq 0\), this interval satisfies the inequality.
- For \(x \in (2, \infty)\), test \(x = 3\):
\[f(3) = \frac{4(3) - 3}{3 - 2} = \frac{9}{1} = 9\]
Since \(9 > 0\), this interval does not satisfy the inequality.
Finally, checking the boundary points: at \(x = \frac{3}{4}\), the expression is \(0\), which satisfies the \(\leq 0\) condition. At \(x = 2\), the expression is undefined. Thus, the solution set is \([\frac{3}{4}, 2)\).
Examples
Rational inequalities are useful in engineering and physics when calculating physical constraints, such as ensuring that the ratio of two dependent variables stays within safe limits. For example, in a heat transfer problem, you might need to determine the range of temperatures (represented by \(x\)) where a specific cooling ratio \(\frac{4x}{x-2}\) is less than or equal to a critical safety threshold \(\frac{3}{x-2}\). Identifying the correct interval ensures that the system operates within safe design parameters.