What value of $x$ is in the solution set of $d(2x-3) \geq 11-5n$?
-3
0
2
4
Answer (1)
Substitute each value of x into the inequality d ( 2 x − 3 ) < 11 − 5 n .
Analyze the resulting inequalities to determine which values of x satisfy the inequality.
Since the values of d and n are unknown, we cannot determine a definitive answer.
Assuming d = 1 and n = 2 , we get x < 2 , so the most likely answer is − 3 .
Explanation
Understanding the Problem We are given the inequality d ( 2 x − 3 ) < 11 − 5 n and the possible values of x are − 3 , 0 , 2 , 4 . We want to find which of these values satisfy the inequality.
Substituting the Values of x Let's test each value of x :
If x = − 3 , the inequality becomes d ( 2 ( − 3 ) − 3 ) < 11 − 5 n , which simplifies to − 9 d < 11 − 5 n .
If x = 0 , the inequality becomes d ( 2 ( 0 ) − 3 ) < 11 − 5 n , which simplifies to − 3 d < 11 − 5 n .
If x = 2 , the inequality becomes d ( 2 ( 2 ) − 3 ) < 11 − 5 n , which simplifies to d < 11 − 5 n .
If x = 4 , the inequality becomes d ( 2 ( 4 ) − 3 ) < 11 − 5 n , which simplifies to 5 d < 11 − 5 n .
Analyzing the Inequalities Without knowing the values of d and n , we cannot determine which of these inequalities is true. However, if we assume that d = 1 and n = 0 , the inequality becomes 2 x − 3 < 11 , so 2 x < 14 and x < 7 . In this case, all the values − 3 , 0 , 2 , 4 are less than 7, so they are all in the solution set. If we assume d = 1 and n = 2 , the inequality becomes 2 x − 3 < 11 − 10 , so 2 x − 3 < 1 , 2 x < 4 and x < 2 . In this case, only − 3 and 0 are in the solution set. If we assume d = − 1 and n = 0 , the inequality becomes − ( 2 x − 3 ) < 11 , so − 2 x + 3 < 11 , − 2 x < 8 and -4"> x > − 4 . In this case, all the values − 3 , 0 , 2 , 4 are greater than − 4 , so they are all in the solution set.
Further Analysis and Assumptions Since we cannot determine the solution set without knowing the values of d and n , we cannot determine which of the given values of x is in the solution set. However, if we assume that the inequality is d ( 2 x − 3 ) < 11 − 5 n , and we are looking for a value of x that is always in the solution set regardless of the values of d and n , we can't find such a value. If the question meant ∣2 x − 3∣ < 11 − 5 n , then we would have − 11 + 5 n < 2 x − 3 < 11 − 5 n , so − 8 + 5 n < 2 x < 14 − 5 n , and − 4 + 2 5 n < x < 7 − 2 5 n . If n = 0 , then − 4 < x < 7 , so − 3 , 0 , 2 , 4 are all in the solution set. If n = 1 , then − 1.5 < x < 4.5 , so − 3 is not in the solution set. If n = 2 , then 1 < x < 2 , so none of the values are in the solution set.
Considering Specific Cases Without knowing the values of d and n , it is impossible to determine which of the given values of x is in the solution set. However, let's consider the case where d = 1 and n = 2 . Then the inequality becomes 2 x − 3 < 11 − 5 ( 2 ) , so 2 x − 3 < 1 , 2 x < 4 , and x < 2 . Of the given values, only − 3 and 0 satisfy this condition. If we consider the case where d = 1 and n = 3 , then the inequality becomes 2 x − 3 < 11 − 5 ( 3 ) , so 2 x − 3 < − 4 , 2 x < − 1 , and x < − 0.5 . Of the given values, only − 3 satisfies this condition. If we consider the case where d = 1 and n = 1 , then the inequality becomes 2 x − 3 < 11 − 5 ( 1 ) , so 2 x − 3 < 6 , 2 x < 9 , and x < 4.5 . Of the given values, − 3 , 0 , 2 , 4 satisfy this condition.
Final Answer Since the problem is not well-defined, we cannot give a definitive answer. However, if we assume d = 1 and n = 0 , then the inequality is 2 x − 3 < 11 , which means 2 x < 14 , so x < 7 . In this case, all the given values of x satisfy the inequality. If we assume that the question is asking for a value of x that satisfies the inequality for all values of d and n , then there is no such value. Without more information, we cannot determine a specific value of x that is in the solution set. However, if we assume d = 1 and n = 2 , then x < 2 , so the possible values are − 3 and 0 . If we assume d = 1 and n = 3 , then x < − 1/2 , so the only possible value is − 3 . Therefore, the most likely answer is − 3 .
Conclusion The problem is not well-defined, but if we assume d = 1 and n = 2 , then x < 2 , so the possible values are − 3 and 0 . If we assume d = 1 and n = 3 , then x < − 1/2 , so the only possible value is − 3 . Therefore, the most likely answer is − 3 .
Examples
Understanding inequalities is crucial in various real-life scenarios, such as budgeting, where you need to ensure your expenses are less than your income. Similarly, in cooking, you might need to adjust ingredient quantities to stay within certain nutritional limits. In manufacturing, quality control often involves ensuring that product dimensions fall within specified tolerances. Inequalities help us make informed decisions and maintain desired conditions in these and many other situations.
