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:

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

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 pesos) of producing $x$ electronic components in a factory is given by the expression $C(x) = \frac{48\,000+900x}{x}$. Determine for which production levels the average cost will be strictly less than $1\,500$ pesos 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?

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 meters) 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:

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