(x-1)^2 + (x+2)^3 >= 0
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To solve the inequality (x-1)^2 + (x+2)^3 >= 0, we can follow these steps:
1. Expand the equation: Expand (x-1)^2 and (x+2)^3 to simplify the expression.
(x-1)^2 = (x-1)(x-1) = x^2 - 2x + 1
(x+2)^3 = (x+2)(x+2)(x+2) = x^3 + 6x^2 + 12x + 8
2. Substitute the expanded expressions back into the original inequality:
x^2 - 2x + 1 + x^3 + 6x^2 + 12x + 8 >= 0
3. Combine like terms:
x^3 + 7x^2 + 10x + 9 >= 0
Now, we need to find the values of x that satisfy this inequality.
4. Factorize the expression (if possible):
The expression x^3 + 7x^2 + 10x + 9 cannot be easily factored.
5. Solve for x:
Since factoring isn't straightforward, let's try using a graphical method. Graph the expression y = x^3 + 7x^2 + 10x + 9 and find the intervals where y is greater than or equal to zero.
By graphing the equation on a graphing calculator or software, we can determine the values of x where y is greater than or equal to zero.
Alternatively, we can look for critical points and analyze the intervals between them to determine where the inequality is satisfied.
6. Analyze the solution intervals:
From the graph or the analysis of intervals, we can determine the intervals where the inequality (x-1)^2 + (x+2)^3 >= 0 is satisfied. These intervals are the solutions to the inequality.
For example, if the graph shows that the inequality is satisfied for x < -3 and -1 < x < 0, then the solution could be written as:
Solution: x < -3 or -1 < x < 0.
Note: The exact intervals may vary depending on the graph or analysis. So, it's important to use a graphing tool or software for accurate results.