Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.
[tex]\frac{x-1}{x-2} \leq 0[/tex]
Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is [tex](-\infty, 1) \cup(2, \infty)[/tex].
(Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
B. The solution set is the empty set.
Answer (2)
The solution set for the inequality x − 2 x − 1 ≤ 0 is [ 1 , 2 ) . This means we include 1, exclude 2, and the values in between where the inequality holds true. Therefore, the correct option is not provided in the choices offered, but the solution is clearly expressed in interval notation.
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Find the critical values by setting the numerator and denominator of the rational expression to zero: x = 1 and x = 2 .
Test the intervals ( − ∞ , 1 ) , ( 1 , 2 ) , and ( 2 , ∞ ) to determine where the inequality x − 2 x − 1 ≤ 0 holds.
Include x = 1 in the solution since the inequality is non-strict ( ≤ 0 ) and exclude x = 2 since the expression is undefined there.
Express the solution set in interval notation: [ 1 , 2 ) .
Explanation
Understanding the Problem We are given the rational inequality x − 2 x − 1 ≤ 0 . Our goal is to find the solution set in interval notation. To do this, we need to consider the critical values where the numerator or denominator is zero.
Finding Critical Values First, we find the critical values by setting the numerator and denominator equal to zero:
Numerator: x − 1 = 0 , which gives x = 1 .
Denominator: x − 2 = 0 , which gives x = 2 .
Creating Intervals Now, we create a number line and mark the critical values 1 and 2 . These critical values divide the number line into three intervals: ( − ∞ , 1 ) , ( 1 , 2 ) , and ( 2 , ∞ ) . We need to test each interval to determine where the inequality x − 2 x − 1 ≤ 0 is satisfied.
Testing Intervals We test each interval:
For the interval ( − ∞ , 1 ) , we choose a test value, say x = 0 . Then 0"> 0 − 2 0 − 1 = − 2 − 1 = 2 1 > 0 . So, this interval is not part of the solution.
For the interval ( 1 , 2 ) , we choose a test value, say x = 1.5 . Then 1.5 − 2 1.5 − 1 = − 0.5 0.5 = − 1 ≤ 0 . So, this interval is part of the solution.
For the interval ( 2 , ∞ ) , we choose a test value, say x = 3 . Then 0"> 3 − 2 3 − 1 = 1 2 = 2 > 0 . So, this interval is not part of the solution.
Determining the Solution Set Since the inequality is ≤ 0 , we include the value x = 1 where the numerator is zero. However, we exclude x = 2 where the denominator is zero because the expression is undefined at x = 2 .
Therefore, the solution set is [ 1 , 2 ) .
Examples
Rational inequalities are useful in various real-world scenarios. For example, they can be used to model the concentration of a drug in the bloodstream over time, where the concentration must remain below a certain threshold to avoid toxicity. They also appear in economics when analyzing cost-benefit ratios, ensuring that the benefits outweigh the costs. In engineering, rational inequalities can help determine the stability of a system, ensuring that certain parameters stay within acceptable limits to prevent failure. Understanding how to solve these inequalities allows us to make informed decisions and maintain safe and efficient operations in many practical situations.
