Solve the following inequation : $\frac{x - 2}{x + 5} > 2$.

Step 1: Rearrange the inequation

To solve the inequation, we first move all terms to one side to compare the expression with zero. This allows us to analyze the sign of the expression.

Step 2: Combine terms into a single fraction

Next, we combine the terms on the left side into a single fraction by finding a common denominator, which is $x + 5$. We then simplify the numerator.

Step 3: Simplify the numerator

We expand the term $2(x+5)$ in the numerator and combine like terms. This simplifies the numerator to $-x - 12$.

Step 4: Multiply by -1 and reverse inequality

To make the leading coefficient of $x$ in the numerator positive, we multiply both the numerator and the right side of the inequality by $-1$. Remember to reverse the inequality sign when multiplying or dividing by a negative number.

Step 5: Determine critical points and intervals

The critical points are the values of $x$ that make the numerator or the denominator equal to zero. These points divide the number line into intervals where the expression's sign can be analyzed. The critical points are $x = -12$ (from $x+12=0$) and $x = -5$ (from $x+5=0$).

Step 6: Analyze the sign of the expression

We test values in the intervals $(-\infty, -12)$, $(-12, -5)$, and $(-5, \infty)$. For $x < -12$ (e.g., $x = -13$): $\frac{-13 + 12}{-13 + 5} = \frac{-1}{ -8} = \frac{1}{8} > 0$. For $-12 < x < -5$ (e.g., $x = -10$): $\frac{-10 + 12}{-10 + 5} = \frac{2}{-5} < 0$. For $x > -5$ (e.g., $x = 0$): $\frac{0 + 12}{0 + 5} = \frac{12}{5} > 0$. We are looking for where $\frac{x + 12}{x + 5} < 0$, which is the interval $(-12, -5)$.

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