What is the solution of $|2 x+7|>27$?
A. $x>-17$ or $x<10$
B. $x<-34$ or $x>20$
C. $x<-17$ or $x>10$
D. $x<-10$ or $x>10$
Answer (1)
Split the absolute value inequality into two cases: 27"> 2 x + 7 > 27 and 2 x + 7 < − 27 .
Solve the first case: 27"> 2 x + 7 > 27 which simplifies to 10"> x > 10 .
Solve the second case: 2 x + 7 < − 27 which simplifies to x < − 17 .
Combine the solutions: The solution is x < − 17 or 10"> x > 10 , so the final answer is 10}"> x < − 17 or x > 10 .
Explanation
Understanding the Problem We are given the absolute value inequality 27"> ∣2 x + 7∣ > 27 . To solve this, we need to consider two separate cases.
Solving Case 1 Case 1: 27"> 2 x + 7 > 27 . We solve for x :
Subtract 7 from both sides: 27 - 7"> 2 x > 27 − 7
20"> 2 x > 20
Divide both sides by 2: \frac{20}{2}"> x > 2 20 10"> x > 10
Solving Case 2 Case 2: 2 x + 7 < − 27 . We solve for x :
Subtract 7 from both sides: 2 x < − 27 − 7 2 x < − 34
Divide both sides by 2: x < 2 − 34 x < − 17
Combining the Solutions Combining the solutions from both cases, we have 10"> x > 10 or x < − 17 .
Final Answer Therefore, the solution to the inequality 27"> ∣2 x + 7∣ > 27 is x < − 17 or 10"> x > 10 .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, consider a machine that produces bolts with a target diameter of 10 mm. Due to manufacturing tolerances, the actual diameter may vary slightly. If we want to ensure that the diameter is within a certain acceptable range, say no more than 0.5 mm from the target, we can express this as an absolute value inequality: ∣ d − 10∣ < 0.5 , where d is the actual diameter. Solving this inequality helps us determine the acceptable range of bolt diameters. Similarly, in finance, absolute value inequalities can be used to model risk tolerance or acceptable deviations from investment goals.
