Exercises: Quadratic inequalities

Solve the following quadratic inequality and express the solution in interval notation:

$x^2 - 5x + 6 \ge 0$

Find the solution set of the inequality:

$-x^2 + 4x + 5 > 0$

Solve the incomplete quadratic inequality:

$2x^2 - 8x \le 0$

Determine the values of $x$ that satisfy the following inequality:

$3x^2 - 27 < 0$

Find the solution set of the inequality:

$x^2 + 6x + 9 \le 0$

State the solution set of the inequality, giving reasons for your answer:

$x^2 - 2x + 5 > 0$

Solve the following system of quadratic inequalities in one variable:

$x^2 - 4 \le 0$

$x^2 - x \ge 0$

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

$4x^2 - 12x + 9 > 0$

The height $h$ in meters of a projectile launched vertically upward as a function of time $t$ in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Set up and solve the corresponding inequality to determine during what time interval the projectile is at a height strictly greater than $17$ meters.

The daily profit of a company, expressed in hundreds of pesos, when manufacturing $x$ units of a product is modeled by the quadratic function $B(x) = -10x^2 + 800x - 7\,000$. How many units must it manufacture daily in order not to make a loss? Express the result as an interval.

A $40$-meter rope is available to fence off a rectangular plot, using an existing wall as one of the sides. If $x$ is the length of the sides perpendicular to the wall, for what values of $x$ will the enclosed area be at least $150 \text{ m}^2$? Note that the dimensions of the plot must be positive.

Solve the following inequality by first expanding and simplifying it to its general form, and express the result as an interval:

$(2x - 1)^2 - (x + 3)(x - 3) < 5x + 14$

From the graphical representation of the parabola $y = f(x)$, visually determine the solution set of the inequality $f(x) \le 0$.

Find the domain of definition of the following real function of a real variable:

$f(x) = \sqrt{-2x^2 + 7x - 3}$

The area of the central rectangular zone of a sports court must be strictly less than $800 \text{ m}^2$. If the length of the court is twice its width $x$ plus $10$ meters, set up the corresponding inequality and deduce what values the width of the court can take.

Solve the following system of inequalities and give the final solution as an interval:

$x^2 - 3x - 10 < 0$

$2x + 5 \ge 1$

Determine for which values of the parameter $m$ the quadratic inequality $x^2 - mx + (m + 3) > 0$ holds for any real number $x$.

The population of a certain invasive species in a lake, measured in thousands of individuals, follows the model $P(t) = -0.2t^2 + 3.2t + 4$, where $t$ is the time in months since the invasion was detected. The authorities decide to intervene when the population exceeds $14,000$ individuals. However, the ecological treatment is only effective if it can be applied for at least $3$ consecutive months during which the population remains above this threshold. Set up the inequality and determine whether it will be possible to successfully apply the treatment according to this model.

An open-topped tray is to be constructed from a rectangular sheet of aluminum measuring $30 \times 20 \text{ cm}$, by cutting out identical squares of side $x$ from the four corners and folding up the resulting flaps. If design requirements dictate that the base area of the tray must be at least $200 \text{ cm}^2$, set up and solve the corresponding inequality to find the interval of possible values of $x$.

Solve the following biquadratic inequality using a substitution of variable and express the final solution as a union of intervals:

$x^4 - 13x^2 + 36 \le 0$

The main suspension cable of a bridge takes the shape of a parabola. Taking the lowest point of the cable as the origin of coordinates, the height $y$ (in meters) of the cable above the roadway at a horizontal distance $x$ (in meters) from the center is given by the function $y = 0.005x^2 + 2$. For aviation safety reasons, obstruction lights must be installed on the sections of the cable whose height above the roadway is between $10 \text{ m}$ and $22 \text{ m}$. Set up the system of inequalities and calculate the intervals of horizontal distance from the center where these lights will be installed.

Solve the modulus inequality by first transforming it into a system of quadratic inequalities:

$|x^2 - 4x| \le 5$

Find the domain of definition of the following function, which combines restrictions from radicands and denominators:

$g(x) = \frac{1}{\sqrt{x^2 - 4}} + \sqrt{9 - x^2}$

Given the quadratic functions $f(x) = x^2 - 2x - 3$ and $g(x) = -x^2 + 4x + 5$, determine analytically on which interval of the number line $f(x) < g(x)$ holds. Based on the attached graph representing both functions and the area enclosed between them, verify the consistency of your result.