Sure, let's solve the inequality [tex]$\frac{(x+2)\left(x^2-2x+1\right)}{4+3x-x^2} \geq 0.3$[/tex] step by step.
1. Write the inequality in standard form:
[tex]\[\frac{(x+2)(x^2-2x+1)}{4+3x-x^2} \geq 0.3\][/tex]
2. Simplify the numerator:
Notice that [tex]\(x^2 - 2x + 1 = (x - 1)^2\)[/tex].
So, the inequality becomes:
[tex]\[\frac{(x+2)(x-1)^2}{4+3x-x^2} \geq 0.3\][/tex]
3. Bring everything to one side of the inequality and set it to zero:
[tex]\[\frac{(x+2)(x-1)^2}{4+3x-x^2} - 0.3 \geq 0\][/tex]
4. Combine the terms over a common denominator:
[tex]\[\frac{(x+2)(x-1)^2 - 0.3(4 + 3x - x^2)}{4 + 3x - x^2} \geq 0\][/tex]
Let's simplify the numerator [tex]\((x+2)(x-1)^2 - 0.3(4 + 3x - x^2)\)[/tex]:
[tex]\[(x+2)(x^2 - 2x + 1) - 0.3(4 + 3x - x^2)\][/tex]
Expanding [tex]\((x+2)(x^2 - 2x + 1)\)[/tex]:
[tex]\[(x+2)(x^2 - 2x + 1) = x^3 - 2x^2 + x + 2x^2 - 4x + 2 = x^3 - 3x + 2\][/tex]
Subtracting the term involving 0.3:
[tex]\[ x^3 - 3x + 2 - (0.3 \times 4 + 0.3 \times 3x - 0.3 \times x^2) = x^3 - 3x + 2 - (1.2 + 0.9x - 0.3x^2)\][/tex]
Simplifying:
[tex]\[ x^3 - 3x + 2 - 1.2 - 0.9x + 0.3x^2 \][/tex]
Combining like terms:
[tex]\[ x^3 + 0.3x^2 - 3.9x + 0.8 \][/tex]
So the inequality is:
[tex]\[\frac{x^3 + 0.3x^2 - 3.9x + 0.8}{4 + 3x - x^2} \geq 0\][/tex]
5. Determine intervals and test points:
Breakdown the domain of [tex]\( x \)[/tex] by finding the critical points of the numerator and the denominator. The intervals created by these critical points will be checked for the sign of the function.
6. Identify and combine the intervals where the function is non-negative:
Through solving and testing the intervals, we find that the solution to the inequality is:
[tex]\[
(1.70856701978624 \leq x < 4.0) \cup (-1.0 < x \leq 0.210958853261712) \cup (-\infty < x \leq -2.21952587304795)
\][/tex]
7. Combine the intervals:
Therefore, the solution to the inequality is:
[tex]\[
\boxed{(1.70856701978624 \leq x < 4.0) \cup (-1.0 < x \leq 0.210958853261712) \cup (-\infty < x \leq -2.21952587304795)}
\][/tex]
