Exercises: Quadratic inequalities

Solve the following quadratic inequality and express the solution as an interval:

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

Find the solution set of the inequality:

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

Solve the incomplete 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, explaining your reasoning:

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

Solve the following system of quadratic inequalities with one unknown:

$x^2 - 4 \le 0$

$x^2 - x \ge 0$

Determine for which values of $x$ the following inequality is satisfied:

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

The height $h$ in feet 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 which time interval the projectile is at a height strictly greater than $17$ feet.

The daily profit of a company, expressed in hundreds of dollars, when manufacturing $x$ units of a product is modeled by the quadratic function $B(x) = -0,5x^2 + 40x - 350$. How many units must it manufacture daily to avoid making a loss? Express the result as an interval.

A rope of $40$ feet is available to fence a rectangular vegetable garden, 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 fenced area be at least $150 \text{ ft}^2$? Keep in mind that the dimensions of the vegetable garden must be positive.

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

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

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

Find the domain of the following real function:

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

The area of the central rectangular zone of a sports court must be strictly less than $800 \text{ ft}^2$. If the length of the court is twice its width $x$ plus $10$ feet, 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 what values of the parameter $m$ the quadratic inequality $x^2 - mx + (m + 3) > 0$ is satisfied 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 apply the treatment successfully according to this model.

A tray without a lid is to be made from a rectangular aluminum sheet of $30 \times 20 \text{ cm}$, by cutting out identical squares of side $x$ at the four corners and folding the resulting flaps upward. If design requirements dictate that the area of the base 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 for $x$.

Solve the following biquadratic inequality using a change 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 feet) of the cable with respect to the roadway at a horizontal distance $x$ (in feet) from the center is given by the function $y = 0,005x^2 + 2$. For aviation safety reasons, clearance lights must be installed on the sections of the cable whose height with respect to the roadway is between $10 \text{ ft}$ and $22 \text{ ft}$. Set up the system of inequalities and calculate in which intervals of horizontal distance from the center these lights will be installed.

Solve the absolute value 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 of 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 in which interval of the real line it is true that $f(x) < g(x)$. Based on the attached graph representing both functions and the area between them, check the consistency of your result.