Rational inequalities

Solve the following rational inequality, expressing its solution as an interval:

$\frac{x-4}{x+3} > 0$

Solve the inequality:

$\frac{x}{x-5} < 0$

Cian tries to solve the rational inequality $\frac{x - 3}{x + 2} \le 0$ on the blackboard and follows this procedure:

"I multiply both sides by $(x + 2)$ to eliminate the denominator: $(x - 3) \le 0 \times (x + 2) \implies x - 3 \le 0 \implies x \le 3$. Therefore, the solution set is $(-\infty, 3]$".

1. a) Identify and explain, with reasoning, the conceptual error that Cian makes when multiplying by $(x + 2)$. Why is this operation not valid in an inequality if we do not know the sign of the expression?
2. b) Correct the exercise by solving the inequality step by step using a sign chart on the corresponding intervals and write the correct solution in interval notation, justifying why one endpoint is open and the other is closed.

Determine the values of $x$ that satisfy:

$\frac{4}{x-6} \le 0$

Solve the following inequality:

$\frac{x^2-9}{x+1} > 0$

Solve the inequality:

$\frac{-5}{2x+3} > 0$

Solve the following rational inequality and express the solution using intervals:

$\frac{x^2-5x+6}{x^2-1} \le 0$

Solve the rational inequality, paying special attention to multiple roots of the numerator:

$\frac{x^2-6x+9}{x+4} \le 0$

Solve the following rational inequality in which the numerator is a cubic polynomial:

$\frac{x^3-8}{x^2-2x-3} \ge 0$

Find the solution set of the inequality:

$\frac{x+3}{x-2} \ge 2$

Determine the set of values that satisfy the following inequality between algebraic fractions:

$\frac{1}{x+1} < \frac{1}{x-3}$

To solve the rational inequality $\frac{2x - 5}{x - 1} \le 1$, three students propose different strategies:

• Aoife subtracts $1$ from both sides, finds a common denominator to obtain a single irreducible fraction $\frac{P(x)}{Q(x)} \le 0$, and constructs the sign chart.

• Seán multiplies both sides by $(x - 1)^2$, stating that since it is a square it is strictly positive for all $x \ne 1$, which avoids reversing the inequality sign and eliminates the denominators.

• Niamh divides the problem into two cases according to the sign of the denominator: Case 1 ($x - 1 > 0$) where she multiplies directly keeping the inequality sign, and Case 2 ($x - 1 < 0$) where she reverses the inequality sign, taking the union of the solutions at the end.

1. a) Analyze the mathematical validity of the three procedures: are they all correct from an algebraic point of view?
2. b) Compare the three methods and justify which one you consider most efficient and reliable to minimize calculation errors in 5th Year.
3. c) Solve the inequality by applying the method you selected as preferable.

The average daily cost (in euros) of producing $x$ electronic components in a factory is given by the expression $C(x) = \frac{800+15x}{x}$. Determine for which production levels the average cost will be strictly less than $25$ euros per component.

The time $T$ (in hours) it takes a boat to travel a distance of $120$ km downstream is modelled by $T(v) = \frac{120}{v+4}$, where $v$ is the boat's own speed in km/h. For which speeds $v > 0$ will the journey take less than $5$ hours?

In optics, the thin lens equation relates the focal length $f$, the object distance $p$ and the image distance $q$ by the formula $\frac{1}{f} = \frac{1}{p} + \frac{1}{q}$. If a lens with a focal length of $f = 12$ cm is used, determine for which object distances $p > 0$ the image distance $q$ will be greater than $24$ cm.

Construct a rational inequality of the form $\frac{P(x)}{Q(x)} \ge 0$, where both $P(x)$ and $Q(x)$ are polynomials of degree 1 or 2 factored into linear factors, such that its solution set is exactly $(-\infty, -3) \cup [2, 5)$.

To justify your construction:

1. Explain in which position (numerator or denominator) you must place each root to ensure which endpoints are included and which are necessarily excluded from the solution set.

2. Determine the overall sign of the fraction (or of the leading coefficient) so that the positive intervals coincide with the requested intervals.

3. Verify the result by constructing the complete sign chart for the inequality you proposed.

Solve the following system of rational inequalities and express the result as a union of intervals:

$\begin{cases} \frac{x-2}{x+3} \ge 0 \\ \frac{x^2-16}{x} \le 0 \end{cases}$

Consider the family of rational inequalities depending on the real parameter $k$:

1. a) Perform a preliminary analysis: study the discriminant of the numerator, $\Delta = k^2 - 64$. Explain how the sign of $\Delta$ affects the number of real roots of the numerator and the possible sign changes on the real line.
2. b) Determine, with reasoning, the positive value of $k$ for which the solution set of the inequality is exactly $(-\infty, 2) \cup \{4\}$. Justify why an isolated point ($x = 4$) appears as part of the solution instead of a continuous interval.
3. c) Using the value of $k$ obtained in the previous section, verify the validity of the result by constructing the sign chart on each of the intervals determined by the roots.
4. d) Without performing additional calculations, deduce what the solution set would be if the original inequality had been strict: $\frac{x^2 - kx + 16}{x - 2} < 0$.