Exercises: Quadratic inequalities

Solve the inequality $x^2 - 5x + 6 \leq 0$ by factorising the polynomial and studying the sign of the factors. Express the solution as an interval.

Solve the inequality $x^2 > 4$ and express the solution using intervals.

Solve the inequality $x^2 + 3x < 0$.

Solve the inequality $(x - 1)(x + 3) > 0$ by constructing a sign table.

Solve the inequality $2x^2 - 8 \geq 0$.

Solve the inequality $x(x - 4) \leq 0$.

The figure shows the parabola $y = x^2 - 2x - 3$.

Use it to solve the inequality $x^2 - 2x - 3 \leq 0$, explaining which region of the graph you need to look at.

Decide whether the following statement is true or false: "the solution to the inequality $x^2 < 9$ is $x < 3$". Justify your answer and give the correct solution.

Lucía has to solve the inequality $x^2 - 6x \geq 0$. On her exam, she writes the following procedure:

"I move the term $6x$ to the right-hand side by adding: $x^2 \geq 6x$.

I divide both sides by $x$: $\frac{x^2}{x} \geq \frac{6x}{x}$.

Therefore, the final solution is $x \geq 6$, which corresponds to the interval $[6, +\infty)$".

1. a) Explain in detail what conceptual error Lucía has made by dividing both sides by $x$ and why that operation is forbidden when we do not know the sign of the unknown.
2. b) Check with a negative value (for example, $x = -2$) that the original inequality is indeed satisfied, thus showing that Lucía's solution is incomplete.
3. c) Solve the inequality correctly by factoring the polynomial, creating the sign table by intervals, and expressing the solution set as a union of intervals.

A small agricultural cooperative in Cieza (Murcia) sells $5\text{ kg}$ boxes of peaches directly to the public. The net weekly profit obtained, $B(x)$ in hundreds of euros, depends on the sale price set for each box, $x$ (in euros), according to the function:

The cooperative needs to set a unit price that guarantees not incurring losses, that is, one that satisfies the inequality $B(x) \geq 0$.

1. a) Set up the inequality, find the roots of the associated equation, and construct the sign table to determine in which price interval the cooperative makes a profit or at least does not lose money.
2. b) From the intervals found, indicate with justification whether the cooperative would have losses or profits selling the box at €$3$ and selling it at €$12$.
3. c) Explain what it means in the real market that, for very high prices (for example, more than €$11$ per box), the profit becomes negative again despite charging more for each unit.

Solve the inequality $x^2 - 2x - 8 \geq 0$.

Solve the inequality $-x^2 + 6x - 5 > 0$. What particularity does it have compared to inequalities where the coefficient of $x^2$ is positive, and how do you solve it?

Solve the inequality $x^2 \leq 3x + 10$.

Solve the system of inequalities $\begin{cases} x^2 - 9 \leq 0 \\ 2x + 1 > 3 \end{cases}$ and express the solution as an interval.

A ball is thrown vertically upwards, and its height, in metres, after $t$ seconds is $h(t) = -5t^2 + 20t$, as shown in the figure.

1. a) Between which times is the ball above $15$ m? b) How long does the full flight of the ball last?

A rectangular plot has a length that is $5$ m longer than twice its width. If its area does not exceed $150\ \text{m}^2$, what values can the width of the plot take?

1. a) Solve the inequality $(x - 2)^2 + 3x \geq x^2 + 2$. b) Although $x^2$ appears in the statement, the inequality is not quadratic: explain why and comment on whether it was necessary to use the method for quadratic inequalities.

A student solves the inequality $x^2 - 4x > 0$ as follows: "I divide both sides by $x$ and get $x - 4 > 0$, so the solution is $x > 4$". Explain the two errors made (one related to the sign of $x$ and another to lost solutions) and solve the inequality correctly.

An environmental technical team monitors water quality in the ría de Huelva using a buoy with a thermal sensor. The safety protocol requires that an acoustic alarm signal be activated only when the water temperature $T$ (in $^{\circ}\text{C}$) satisfies a quadratic inequality.

We know that the alarm remains off (safe temperature range) exactly in the open interval $(12, 20)$, and turns on both if the water drops to $12\text{ }^{\circ}\text{C}$ or lower, and if it rises to $20\text{ }^{\circ}\text{C}$ or higher.

1. a) Write a quadratic inequality of the form $T^2 + bT + c \geq 0$ whose solution is exactly the set of temperatures at which the alarm is activated.
2. b) If a more demanding new protocol requires the alarm to be activated when the temperature belongs to the closed interval $[14, 18]$, write an inequality of the form $-T^2 + b'T + c' \geq 0$ that models this new condition.
3. c) Justify how the endpoints of the intervals are related to the solutions of the associated quadratic equation and why the direction of the inequality changes depending on the sign of the leading coefficient.

According to road safety studies used by the Dirección General de Tráfico (DGT), the total stopping distance of a passenger car on dry asphalt and with tyres in good condition (adding the reaction distance and the mechanical braking distance), measured in metres, is approximated according to the speed $v$ (in $\text{km/h}$) by the expression:

On an urban thoroughfare with a speed limit of $50\text{ km/h}$, a driver spots a pedestrian crossing $24$ metres away.

1. a) Make a reasoned estimate before carrying out exact calculations: mentally calculate the approximate braking distance for $40\text{ km/h}$ and for $60\text{ km/h}$. From these two values, deduce whether the maximum speed that would prevent hitting the pedestrian will be above or below the legal limit of $50\text{ km/h}$.
2. b) Set up the quadratic inequality $d(v) < 24$ that determines the speeds at which the vehicle manages to stop in less than $24$ metres. Solve it analytically, taking into account the real physical constraints of speed ($v \geq 0$).
3. c) Check whether the speed interval obtained confirms your previous estimate and write a brief conclusion about the danger of driving at the maximum permitted speed ($50\text{ km/h}$) compared to driving at $30\text{ km/h}$ in streets with reduced visibility.

In a fireworks display, the height $h$, in metres, of a rocket $t$ seconds after launch is given by $h(t) = -5t^2 + 40t$. a) At what times is the rocket exactly at a height of $60$ m? b) For how long is it above $60$ m? c) How long does the full flight of the rocket last?

Find the values of $k$ for which the inequality $x^2 + kx + 9 > 0$ holds for any real number $x$. Hint: think about how many real roots the polynomial must have for this to happen.

Solve the inequality $x^4 - 13x^2 + 36 \leq 0$ using the substitution $z = x^2$. Express the solution as a union of intervals.

Solve the inequality $(x + 1)(x - 2)(x - 4) \geq 0$ by constructing a sign table with the three roots.

Solve the inequality $\frac{x - 1}{x + 2} > 0$ by studying the signs of the numerator and denominator. Explain why $x = -2$ could not be part of the solution even if the inequality were non-strict.

The weekly profit, in euros, that a bicycle repair shop makes when repairing $x$ bikes is $B(x) = -2x^2 + 80x$. a) For what number of weekly repairs does the shop make a profit? b) For which ones does the profit exceed $600\ \text{€}$? c) How many weekly repairs maximise the profit?

1. a) Solve the two inequalities in the system $\begin{cases} x^2 - 6x + 5 \leq 0 \\ x > 6 \end{cases}$ separately. b) The figure shows the parabola $y = x^2 - 6x + 5$ and the vertical line $x = 6$: use it to interpret the result. Does the system have a solution? Why or why not?

It is intended to use $40$ m of fencing to enclose a rectangular pen using an existing wall as one of the long sides, as shown in the figure.

1. a) Express the area of the pen as a function of the width $x$. b) For what values of $x$ does the area exceed $150\ \text{m}^2$? c) What is the maximum area that can be achieved, and with what width is it reached?

In a technical study on the strength of aluminium profiles, two quadratic inequalities present particular cases:

• Inequality 1: $x^2 - 8x + 16 > 0$

• Inequality 2: $-x^2 + 2x - 5 < 0$

Three procedures can be applied to solve them:

• Method A: Find the roots using the quadratic formula, factor the polynomial, and construct a sign table by intervals on the real line.

• Method B: Calculate the discriminant $\Delta = b^2 - 4ac$ and interpret the position of the parabola with respect to the $X$-axis according to the sign of the coefficient $a$.

• Method C: Recognize special products (identities) or complete the square to analyze the sign of the expression analytically.

1. a) For Inequality 1, justify why Method C is much more efficient and reliable than Method A. Apply Method C to solve it and state its solution set accurately.
2. b) For Inequality 2, justify why Method B allows you to give the solution almost immediately. Carry out the corresponding reasoning and write the solution set.
3. c) Explain in detail why Method A (the classical sign table) often leads students to make errors when $\Delta \leq 0$, and argue what conceptual advantage is gained by visualizing the inequality as the position of a parabola relative to the $X$-axis.

A logistics warehouse located in Antequera (Málaga) plans to install photovoltaic panels on its roof for industrial self-consumption. The net monthly economic savings $A(n)$, in euros, when installing $n$ solar panels is given by the quadratic model:

(in which the monthly fee for depreciation and maintenance of the installation is already deducted).

Management imposes two simultaneous constraints to approve the project:

1. Profitability: the net monthly savings must be at least €$600$ ($A(n) \geq 600$).

2. Structural safety: the maximum weight supported by the roof limits the total number of panels to a maximum of $35$ units ($n \leq 35$).

1. a) Set up the inequality associated with the profitability condition, transform it to its general form with zero on one side, and find the continuous interval of real values of $n$ that satisfy $A(n) \geq 600$.
2. b) Taking into account that the number of panels must be an integer and combining profitability with the roof limitation, determine the allowable range of panels $[n_{\min}, n_{\max}]$ that the company can decide to install.
3. c) Calculate the monthly savings obtained by installing the minimum allowable number of panels ($n_{\min}$) and compare it with the savings obtained by installing the maximum allowable number of $35$ panels. Based on the shape of the parabola, provide a reasoned justification for why adding more panels does not always produce greater economic savings beyond a certain threshold.

Unit Challenge. How can you optimize electricity consumption in your home to pay just the right amount? Become an energy auditor using real data from Red Eléctrica de España (REE).

1. Data and linear model: Check today's average $\text{kWh}$ price on the REE website (esios.ree.es) (e.g. $0,14$ €/$\text{kWh}$) and the fixed charge on your bill. Set up a linear equation to find the consumption that results in an exact cost of $45$ €, and a linear inequality to not exceed that monthly limit.

2. Space and quadratic equations: Plan the installation of solar panels on a rectangular surface of width $x$ and length $x+3$. Solve a quadratic inequality so that the occupied surface area does not exceed $18\text{ m}^2$ ($x(x+3) \le 18$) and a quadratic equation to set a structure cost of $20x^2 + 10x = 300$.

3. Average cost and report: Set up a rational equation to calculate the average cost per $\text{kWh}$ according to consumption. Graph the solutions in GeoGebra or in a spreadsheet and write a reasoned one-page proposal for your family.

Assessment criteria:

• Correct setup and solving of equations and inequalities.

• Rigorous integration of real data from REE and contextualized verification of solutions.

• Clarity of the final proposal and support from GeoGebra or the spreadsheet.