Answer:
Explanation:
To solve the equation, we'll need to consider two cases:
Case 1: 2x - 3 is positive or zero
If 2x - 3 is positive or zero, then |2x - 3| = 2x - 3, and the inequality becomes:
2x - 3 < 3 + x - x^2
Rearranging and simplifying:
x^2 - x - 6 < 0
Factoring:
(x - 3)(x + 2) < 0
The solutions to this inequality are:
-2 < x < 3
However, we still need to check that 2x - 3 is indeed positive or zero for this range of x. We can see that this is true for x in the range (-2, 3), so this is a valid solution.
Case 2: 2x - 3 is negative
If 2x - 3 is negative, then |2x - 3| = -(2x - 3), and the inequality becomes:
-(2x - 3) < 3 + x - x^2
Rearranging and simplifying:
x^2 - 3x - 6 < 0
Factoring:
(x - 3)(x + 2) > 0
The solutions to this inequality are:
x < -2 or x > 3
However, we still need to check that 2x - 3 is indeed negative for this range of x. We can see that this is true for x < -2, so this is a valid solution.
Putting these two cases together, we get the solution:
x < -2 or -2 < x < 3
I hope this helps! Let me know if you have any questions.