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Q.
- Set Inequality Equal Zero: First, let's set the inequality to equal zero to find the critical points.
- Solve for x: Now, we need to solve for x. Let's isolate .
- Exponentiate Both Sides: To get rid of the natural log, we'll exponentiate both sides with base .
- Find Approximate Solution: Since is just , we have:
- Test Critical Points: This equation isn't easy to solve algebraically, so we'll look at it graphically or use numerical methods to find the approximate value of .
- Test : By graphing or using numerical methods, we find that the approximate solution is
- Test : Now we need to test intervals around our critical point to see where the inequality holds true.
- Final Solution: Let's test a value less than , say . We get ext{ln}(1) - rac{1}{1} = 0 - 1, which is less than . So the inequality doesn't hold for .
- Final Solution: Let's test a value less than , say . We get ext{ln}(1) - rac{1}{1} = 0 - 1, which is less than . So the inequality doesn't hold for .Now let's test a value greater than , say . We get ext{ln}(2) - rac{1}{2}, which is greater than . So the inequality holds for .
- Final Solution: Let's test a value less than , say . We get ext{ln}(1) - rac{1}{1} = 0 - 1, which is less than . So the inequality doesn't hold for . Now let's test a value greater than , say . We get ext{ln}(2) - rac{1}{2}, which is greater than . So the inequality holds for . Therefore, the solution to the inequality is .
