Answer :
Answer:
The wavy curve method is a graphical approach to solve inequalities. Here's how to solve the given inequality using this method:
1. Write the inequality: x - 2/x + 2 > 2x - 3/4x - 1
2. Simplify the inequality by combining like terms:
(x - 2)/(x + 2) > (8x - 3)/(4x - 1)
1. Graph the related functions:
y = (x - 2)/(x + 2) and y = (8x - 3)/(4x - 1)
1. Identify the critical points (asymptotes and x-intercepts) of both functions:
- (x - 2)/(x + 2) has asymptotes at x = -2 and x-intercept at x = 2
- (8x - 3)/(4x - 1) has asymptotes at x = 1/4 and x-intercept at x = 3/8
1. Draw the wavy curves:
- Start from the left and move right:
- For x < -2, both functions are negative, so the inequality is false (shaded region).
- For -2 < x < 1/4, the first function is positive, and the second is negative, so the inequality is true (unshaded region).
- For 1/4 < x < 2, both functions are positive, so the inequality is false (shaded region).
- For 2 < x, the first function is positive, and the second is negative, so the inequality is true (unshaded region).
1. Determine the solution:
The inequality is true for -2 < x < 1/4 and x > 2. Combining these intervals, the solution is:
(-2, 1/4) ∪ (2, ∞)
Note: The wavy curve method helps visualize the solution, but it's essential to verify the critical points and the solution interval algebraically to ensure accuracy.
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