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:

$3x^2 - 27 < 0$

To solve the quadratic inequality $x^2 \ge 6x$, Cian writes the following reasoning on his exam:

"I divide both sides by $x$ and I get $x \ge 6$. Therefore, the solution set is the interval $[6, +\infty)$".

1. a) Identify and explain in detail the conceptual error made by Cian. Why is it not allowed to divide an inequality by an unknown without previously knowing its sign?
2. b) Correct the exercise by solving the inequality using the general procedure (moving everything to the left-hand side, finding the roots, and studying the sign on the resulting intervals) and express the correct solution set in interval notation.

Construct a quadratic inequality of the form $ax^2 + bx + c < 0$ that simultaneously satisfies two conditions:

1) Its solution set is exactly the open interval $(-4, 1)$.

2) The leading coefficient is $a = 2$.

1. a) Reasonably determine the numerical values of the coefficients $b$ and $c$.
2. b) Explain, using the sign of $a$ and the shape of the associated parabola, why the resulting polynomial takes strictly negative values only in the interior section between both roots and not on the outside.

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 $f(x) \le 0$.

Consider the quadratic inequality $(2x - 1)^2 < 3x + 14$.

1. a) Estimate: Before performing any algebraic transformation, mentally evaluate both sides of the inequality at the integers $x = -2$, $x = -1$, $x = 2$, and $x = 3$. With these data, deduce between which two consecutive integers the smaller root must lie and between which two consecutive integers the larger root must lie. From this, venture a reasoned estimate of the solution interval.
2. b) Solve: Move all terms to the left-hand side, reduce the polynomial to the form $ax^2 + bx + c < 0$, calculate its exact roots using the quadratic formula, and determine the solution set analytically using a sign chart.
3. c) Check: Use the approximation $\sqrt{257} \approx 16.03$ to express the endpoints of the interval in decimal form and check whether your estimate from part a) was consistent. Finally, explain based on the continuity of polynomial functions why the sign of the quadratic polynomial cannot change at any intermediate point between the two calculated roots.

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:

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

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

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

It is proposed to solve the quadratic inequality $4x^2 - 12x + 9 \le 0$ in class. Two students choose different strategies:

• Method 1 (Seán): He applies the general quadratic formula, checks that the discriminant is zero, finds the double root $x = 1.5$, divides the real line into two intervals to the left and right of $1.5$, and evaluates a test number in each one to deduce the sign.

• Method 2 (Aoife): She immediately recognizes that the polynomial is the special product $(2x - 3)^2$ and deduces the solution directly by reasoning about the properties of squares in real numbers.

1. a) Explain how Aoife deduces the solution from the special product without needing to test any numerical value.
2. b) Compare the effectiveness of both methods and justify which one is more advantageous in this case.
3. c) What would have happened if the initial inequality had been $4x^2 - 12x + 9 < 0$? And what if it had been $4x^2 - 12x + 9 > 0$? Write the solution for each of these two new cases, justifying your answer.

The height $h$ in metres of a projectile launched vertically upwards 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 $17$ metres.

Solve the following system of quadratic inequalities in one variable:

$x^2 - 4 \le 0$

$x^2 - x \ge 0$

The daily profit of a company, expressed in hundreds of euros, when manufacturing $x$ units of a product is modelled by the quadratic function $B(x) = -0.5x^2 + 40x - 350$. How many units must it manufacture daily in order not to make a loss? Express the result as an interval.

A $40$-metre rope is available to fence off a rectangular allotment, 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 allotment must be positive.

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 $10$ metres, set up the corresponding inequality and deduce what values the width of the court can take.

During a cold snap in County Cavan, Met Éireann (the Irish Meteorological Service) records that the temperature $T$ (in $^{\circ}\text{C}$) over the course of a winter day depends on the solar hour $t$ (with $t \in [0, 24]$ measured in hours from 00:00) according to the quadratic function:

To prevent the water pipes of a livestock farm from freezing, the automatic heating system must be activated as long as the temperature is strictly negative ($T(t) < 0$).

1. a) Set up the corresponding quadratic inequality and solve it analytically by finding the roots and drawing up the sign chart.
2. b) Express the solution in interval notation respecting the daily time domain, $[0, 24]$.
3. c) Interpret the result obtained, indicating precisely during which time intervals of the day the heating will remain on.

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 aluminium 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 metres) of the cable above the roadway at a horizontal distance $x$ (in metres) from the centre 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 centre 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.

The goal is to uniquely determine the coefficients of a quadratic inequality of the form $ax^2 + bx + c \le 0$ from the following information about the polynomial $P(x) = ax^2 + bx + c$:

1) The solution set of the inequality $P(x) \le 0$ is exactly the closed interval $[-1, 5]$.

2) The vertex of the parabola associated with $P(x)$ reaches an absolute minimum value equal to $-18$.

1. a) Reasonably justify the sign of the leading coefficient $a$ based on the geometry of the parabola and the fact that the solution is a single bounded interval rather than the union of two disjoint rays.
2. b) Find the exact numerical values of $a$, $b$, and $c$.
3. c) Without performing additional algebraic calculations or drawing up new sign charts, deduce with justification the solution set of the following three inequalities:

• $ax^2 + bx + c < -18$

• $ax^2 + bx + c \ge -18$

• $ax^2 + bx + c > 0$

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 $x$.