Exercises: Rational inequalities

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

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

Find the solution set of the inequality:

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

Determine for which values of $x$ the following inequality holds:

$\frac{-x+5}{x+2} \ge 0$

Solve the inequality:

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

Find the solution interval of the rational inequality:

$\frac{3x-9}{-x+4} \ge 0$

Solve the following inequality:

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

Determine the values of $x$ that satisfy:

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

Solve the inequality:

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

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.

Given the rational inequality $\frac{x + 5}{x - 1} > 0$:

1. a) Before performing any formal calculations, estimate the behavior of the fraction: what sign will it have if we evaluate very large positive values of $x$ (for example, $x = 100$)? And if we take negative values with a very large absolute value (for example, $x = -100$)? From this observation, hypothesize whether the solution will consist of the union of two infinite rays or a single bounded central interval.
2. b) Solve the inequality by finding the roots of the numerator and denominator and constructing the sign chart.
3. c) Check whether the result obtained confirms your initial estimation and explain why both endpoints must be open even though the numerator becomes zero at $x = -5$.

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

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

Find the solution set of the inequality:

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

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?

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

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

The growth rate of a bacterial population (in thousands of bacteria per hour) as a function of time $t$ (in hours) is given by the function $R(t) = \frac{t^2-16}{t+2}$. In which time intervals $t \ge 0$ is the growth rate strictly positive?

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

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

The concentration of a drug in the bloodstream, measured in mg/L, $t$ hours after its administration is given by the function $C(t) = \frac{4t}{t^2+4}$. During which time interval $t \ge 0$ is the concentration of the drug greater than or equal to $1$ mg/L?

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.

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.

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

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

Determine the domain of definition of the function of a real variable $f(x) = \sqrt{\frac{x^2-x-2}{x^2+4x+3}}$.

Find the solution set of the following rational inequality:

$\frac{x+2}{x-1} \le \frac{x-4}{x+2}$

Determine for which values of the real parameter $k$ the following inequality holds for all $x \in \mathbb{R}$:

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

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.

Solve the rational inequality, first factorising the polynomials in the numerator and denominator:

$\frac{x^4-16}{x^3-2x^2} \le 0$

The thermal efficiency index $E(r)$ of a cylindrical cooling system depends on the radius $r$ (in metres) of the ducts according to the function $E(r) = \frac{r^2-4r+3}{r^2-6r+8}$. Engineers require this index to be strictly negative so that the system does not overheat. For which values of the radius $r > 0$ is the system safe? Use the graphical representation of the function to justify your answer.

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

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

A group of 1st Year students from Galway is organizing a trip to the Burren. The coach company charges a fixed fee of $540$ € to rent the vehicle plus $12$ € per occupied seat for travel insurance and management. The coach has a maximum capacity of $55$ seats and, according to company policy, does not make the trip if fewer than $15$ people travel.

For the activity to be approved by the school, the average cost per participating student, $C_m(x)$, must not exceed $25$ €.

1. a) Write the analytic expression for the average cost $C_m(x)$ as a function of the number of students $x$ and set up the corresponding rational inequality, stating the restrictions on the variable $x$ imposed by the real-world context.
2. b) Algebraically solve the rational inequality by moving all terms to one side, reducing to a single fraction, and studying the signs of the factors.
3. c) Determine the minimum number of students that must sign up for the field trip to be economically viable.
4. d) If, due to an increase in fuel prices, the company raises the fixed cost to $630$ €, calculate how many additional students would be needed at minimum compared to the previous case in order not to exceed the limit of $25$ € per person.

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: $\frac{x^2 - kx + 16}{x - 2} < 0$.