Key Concepts and Formulas

• Wavy Curve Method: Used to solve rational inequalities by analyzing the sign of the expression in different intervals defined by the roots of the numerator and denominator.
• Quadratic Discriminant: For a quadratic $ax^2 + bx + c$, the discriminant $D = b^2 - 4ac$ determines the nature of the roots. If $a > 0$ and $D < 0$, the quadratic is always positive.
• Exponential Properties: $a^{m+n} = a^m \cdot a^n$ and $a^{mn} = (a^m)^n$. If $a > 1$ and $a^x \le a^y$, then $x \le y$.

Step-by-Step Solution for $S_1$

Step 1: Analyze the given inequality and identify the domain.

• Why: To understand the problem and determine any restrictions on the possible values of $x$.
• The inequality is: $\frac{(x + 2)({x^2} + 3x + 5)}{ - 2 + 3x - {x^2}} \ge 0$
• The domain is $x \in R - \{1, 2\}$, meaning $x \ne 1$ and $x \ne 2$, because these values would make the denominator zero.

Step 2: Analyze the quadratic factor in the numerator.

• Why: To simplify the inequality by identifying factors that are always positive or negative and therefore do not change the sign of the expression.
• The quadratic factor is ${x^2} + 3x + 5$.
• The coefficients are $a = 1$, $b = 3$, and $c = 5$.
• The discriminant is $D = b^2 - 4ac = (3)^2 - 4(1)(5) = 9 - 20 = -11$.
• Since $a = 1 > 0$ and $D = -11 < 0$, the quadratic ${x^2} + 3x + 5$ is always positive for all real values of $x$.

Step 3: Simplify the inequality by removing the positive quadratic factor.

• Why: Since the quadratic factor is always positive, dividing both sides of the inequality by it will not change the inequality's direction.
• The inequality becomes: $\frac{x + 2}{ - 2 + 3x - {x^2}} \ge 0$

Step 4: Factor and standardize the denominator.

• Why: To apply the Wavy Curve Method effectively, it's helpful to have the leading coefficient of each factor be positive.
• Factor out $-1$ from the denominator: $- 2 + 3x - {x^2} = -({x^2} - 3x + 2)$
• Factor the quadratic in the denominator: ${x^2} - 3x + 2 = (x - 1)(x - 2)$
• The denominator is $-(x - 1)(x - 2)$.
• The inequality becomes: $\frac{x + 2}{-(x - 1)(x - 2)} \ge 0$

Step 5: Multiply by -1 and reverse the inequality sign.

• Why: To get a positive leading coefficient in the denominator.
• Multiply both sides by $-1$: $\frac{x + 2}{(x - 1)(x - 2)} \le 0$

Step 6: Apply the Wavy Curve Method.

• Why: To determine the intervals where the expression is less than or equal to zero.
• Identify the critical points: $x = -2$, $x = 1$, and $x = 2$.
• These points divide the number line into the intervals: $(-\infty, -2)$, $(-2, 1)$, $(1, 2)$, and $(2, \infty)$.
• Test the sign of the expression in each interval:
  • $x < -2$: $\frac{(-)}{(-)(-)} = (-)$
  • $-2 < x < 1$: $\frac{(+)}{(-)(-)} = (+)$
  • $1 < x < 2$: $\frac{(+)}{(-)(+)} = (-)$
  • $x > 2$: $\frac{(+)}{(+)(+)} = (+)$
• The inequality requires the expression to be $\le 0$.
• Therefore, the solution is $(-\infty, -2] \cup (1, 2)$.

Step 7: State the solution set for $S_1$.

• Why: To summarize the result of solving the inequality for $S_1$.
• $S_1 = (-\infty, -2] \cup (1, 2)$.

Step-by-Step Solution for $S_2$

Step 1: Rewrite the exponential inequality.

• Why: To prepare the inequality for substitution and simplification.
• The inequality is: ${3^{2x}} - {3^{x + 1}} - {3^{x + 2}} + 27 \le 0$
• Rewrite the terms: $(3^x)^2 - 3 \cdot 3^x - 9 \cdot 3^x + 27 \le 0$

Step 2: Perform a substitution.

• Why: To convert the exponential inequality into a quadratic inequality.
• Let $t = 3^x$. Since $3^x > 0$ for all real $x$, we have $t > 0$.
• The inequality becomes: $t^2 - 3t - 9t + 27 \le 0$

Step 3: Simplify and solve the quadratic inequality.

• Why: To find the range of values for $t$ that satisfy the inequality.
• Combine like terms: $t^2 - 12t + 27 \le 0$
• Factor the quadratic: $(t - 3)(t - 9) \le 0$
• The roots are $t = 3$ and $t = 9$.
• Since the quadratic has a positive leading coefficient, the solution is $3 \le t \le 9$.

Step 4: Substitute back to find the solution for $x$.

• Why: To find the values of $x$ that satisfy the original inequality.
• Replace $t$ with $3^x$: $3 \le 3^x \le 9$
• Rewrite as powers of 3: $3^1 \le 3^x \le 3^2$
• Since the base is greater than 1, we can compare the exponents: $1 \le x \le 2$

Step 5: State the solution set for $S_2$.

• Why: To summarize the result of solving the inequality for $S_2$.
• $S_2 = [1, 2]$.

Step-by-Step Solution for $S_1 \cup S_2$

Step 1: Find the union of the two sets.

• Why: To find all values of $x$ that are in either $S_1$ or $S_2$ (or both).
• $S_1 = (-\infty, -2] \cup (1, 2)$
• $S_2 = [1, 2]$
• $S_1 \cup S_2 = (-\infty, -2] \cup [1, 2]$

Common Mistakes & Tips

• Remember to flip the inequality sign when multiplying or dividing by a negative number.
• Always consider the domain restrictions when dealing with rational expressions.
• When using substitutions, remember to substitute back to the original variable.

Summary

We solved the two inequalities separately to find $S_1 = (-\infty, -2] \cup (1, 2)$ and $S_2 = [1, 2]$. Then, we found the union of these sets to be $S_1 \cup S_2 = (-\infty, -2] \cup [1, 2]$.

The final answer is \boxed{( - \infty , - 2] \cup [1,2]}, which corresponds to option (B).