Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.
[tex]\frac{(8-2 x)}{2 x+3} \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 [ ].
(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 2 x + 3 ( 8 − 2 x ) ≤ 0 is ( − ∞ , − 2 3 ) ∪ [ 4 , ∞ ) . This includes all values less than − 2 3 and all values greater than or equal to 4 .
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Find the critical values by setting the numerator and denominator to zero: x = 4 and x = − 2 3 .
Test the intervals ( − ∞ , − 2 3 ) , ( − 2 3 , 4 ) , and ( 4 , ∞ ) to see where the inequality holds.
The intervals ( − ∞ , − 2 3 ) and ( 4 , ∞ ) satisfy the inequality.
Include x = 4 in the solution, but exclude x = − 2 3 . The solution set is ( − ∞ , − 2 3 ) ∪ [ 4 , ∞ ) .
Explanation
Understanding the Problem We are given the rational inequality 2 x + 3 ( 8 − 2 x ) ≤ 0 . Our goal is to find the solution set for this inequality and express it in interval notation. This means we need to determine the values of x for which the expression is less than or equal to zero.
Finding Critical Values First, we need to find the critical values of x where the expression equals zero or is undefined. This occurs when the numerator is zero or the denominator is zero.
Setting the numerator equal to zero: 8 − 2 x = 0 2 x = 8 x = 4
Setting the denominator equal to zero: 2 x + 3 = 0 2 x = − 3 x = − 2 3
Creating Test Intervals Now we have two critical values: x = 4 and x = − 2 3 . These values divide the number line into three intervals: ( − ∞ , − 2 3 ) , ( − 2 3 , 4 ) , and ( 4 , ∞ ) . We will test a value from each interval to see if it satisfies the inequality.
Testing the Intervals
Interval ( − ∞ , − 2 3 ) : Choose x = − 2 . Then, 2 ( − 2 ) + 3 8 − 2 ( − 2 ) = − 4 + 3 8 + 4 = − 1 12 = − 12 ≤ 0 This interval satisfies the inequality.
Interval ( − 2 3 , 4 ) : Choose x = 0 . Then, 0"> 2 ( 0 ) + 3 8 − 2 ( 0 ) = 3 8 > 0 This interval does not satisfy the inequality.
Interval ( 4 , ∞ ) : Choose x = 5 . Then, 2 ( 5 ) + 3 8 − 2 ( 5 ) = 10 + 3 8 − 10 = 13 − 2 ≤ 0 This interval satisfies the inequality.
Determining the Solution Set Since the inequality is non-strict ( ≤ 0 ), we include the value x = 4 in the solution set because the expression equals zero at this point. However, we exclude x = − 2 3 because the denominator would be zero, making the expression undefined.
Therefore, the solution set is ( − ∞ , − 2 3 ) ∪ [ 4 , ∞ ) .
Final Answer The solution set for the inequality 2 x + 3 ( 8 − 2 x ) ≤ 0 is ( − ∞ , − 2 3 ) ∪ [ 4 , ∞ ) .
Examples
Rational inequalities are useful in various real-world scenarios. For example, a company might use them to determine the production levels needed to maintain a certain profit margin. Suppose the profit margin P ( x ) depends on the number of units produced x and is given by a rational function. The company can solve the inequality P ( x ) ≥ k to find the production levels x that ensure the profit margin is at least k . This helps in making informed decisions about production and pricing strategies.
