Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.
$\frac{-x+5}{x-7} \geq 0$
Answer (1)
Find the critical values by setting the numerator and denominator to zero: x = 5 and x = 7 .
Test the intervals ( − ∞ , 5 ) , ( 5 , 7 ) , and ( 7 , ∞ ) to determine where the inequality holds.
Include x = 5 in the solution since the inequality is ≥ 0 , but exclude x = 7 since the denominator cannot be zero.
Express the solution in interval notation: [ 5 , 7 ) .
Explanation
Problem Analysis We are given the rational inequality x − 7 − x + 5 ≥ 0 . Our goal is to solve this inequality, represent the solution set on a number line, and express it in interval notation.
Finding Critical Values To solve the inequality, we first find the critical values by setting the numerator and denominator equal to zero:
Numerator: − x + 5 = 0 ⇒ x = 5 Denominator: x − 7 = 0 ⇒ x = 7
These critical values divide the number line into three intervals: ( − ∞ , 5 ) , ( 5 , 7 ) , and ( 7 , ∞ ) .
Testing Intervals Now, we test each interval to see where the inequality x − 7 − x + 5 ≥ 0 holds true.
Interval ( − ∞ , 5 ) : Choose x = 0 . Then 0 − 7 − 0 + 5 = − 7 5 = − 7 5 < 0 . The inequality is not satisfied.
Interval ( 5 , 7 ) : Choose x = 6 . Then 0"> 6 − 7 − 6 + 5 = − 1 − 1 = 1 > 0 . The inequality is satisfied.
Interval ( 7 , ∞ ) : Choose x = 8 . Then 8 − 7 − 8 + 5 = 1 − 3 = − 3 < 0 . The inequality is not satisfied.
Considering Endpoints Since the inequality is ≥ 0 , we include the value x = 5 where the numerator is zero, making the fraction equal to zero. However, we must exclude x = 7 because the denominator cannot be zero, as division by zero is undefined.
Expressing the Solution in Interval Notation Therefore, the solution set consists of the interval ( 5 , 7 ) along with the point x = 5 . Combining these, we get the interval [ 5 , 7 ) .
Final Answer The solution set in interval notation is [ 5 , 7 ) . This means that all numbers from 5 (inclusive) up to 7 (exclusive) satisfy the given inequality.
Examples
Understanding rational inequalities is crucial in various real-world scenarios. For instance, consider a business trying to optimize its profit margin. Let's say the profit margin is represented by the inequality x − 7 − x + 5 ≥ 0 , where x is the number of units produced. Solving this inequality helps the business determine the range of production levels that ensure a non-negative profit margin. If producing between 5 and 7 units (but not exactly 7) guarantees a profit, the business can make informed decisions about its production strategy to maximize profitability. This concept extends to other areas like resource allocation, investment analysis, and risk management, where understanding constraints and optimizing outcomes is essential.
