Exercises: Linear and quadratic inequalities

Solve the following first-degree inequality and represent the solution on the real number line using an interval:

$5 - 2x \le 11$

Draw a real number line in your notebook with a suitable scale, mark the endpoint of the interval and indicate whether it is included or not. Then write the solution using interval notation.

A group of friends wants to organise a graduation party. The venue rental costs a fixed $150$ €, and the catering costs $12$ € per attendee. If they have a maximum budget of $600$ €, set up an inequality and calculate the maximum number of people who can attend the party.

Determine the solution set of the following quadratic inequality by analyzing the sign of the factors:

$x^2 - 9 > 0$

Express the result as intervals.

Check whether the values $x = 0$, $x = 2$ and $x = 5$ are solutions of the quadratic inequality:

$x^2 - 5x + 4 \le 0$

Find the solution of the following system of linear inequalities with one unknown and express it as an interval:

$\begin{cases} x + 3 > 1 \\ 2x - 4 \le 2 \end{cases}$

You can use this number line to represent the solutions of each inequality and their intersection:

The area of a rectangle with base $5$ cm and height $(x - 2)$ cm must be less than $30$ cm². What values can the variable $x$ take? Note that the height must always be a positive quantity.

Solve the following inequality with brackets and simplify the result:

$3(x - 1) + 2 > 5x - 7$

A phone company offers a plan charging a fixed monthly fee of $10$ € plus $0,05$ € per minute of call. Another company charges a fixed fee of $5$ € but $0,15$ € per minute. After how many minutes of calls per month does the first company become cheaper?

Lucía solves the linear inequality $7 - 3x \le 19$ to find out above what value an economic variable becomes acceptable. On her exam, she writes the following steps:

• Step 1: $-3x \le 19 - 7$

• Step 2: $-3x \le 12$

• Step 3: $x \le \frac{12}{-3}$

• Step 4: $x \le -4$

Lucía concludes that the solution set is $(-\infty, -4]$. Identify in which step she made an error, explain which algebraic property of inequalities she failed to respect (basing your explanation on what happens to the order on the real line when multiplying or dividing by a negative number), and write the correct solution, stating the final result as an interval.

In a Spanish provincial capital, the Transport Consortium offers two options for urban travel: a monthly pass with a flat fare of $20$ € with unlimited trips, and a 10-trip card costing $6,10$ € (that is, $0,61$ € per individual trip). A 1st-year Bachillerato student wants to know whether it is cost-effective to buy the monthly pass to travel to high school during a school month. Set up a linear inequality expressing the necessary condition for the cost of the monthly pass to be less than the total expense on single tickets in terms of the number of monthly trips, $x$. Solve the inequality and answer with reasoning: if the month has $21$ school days and the student only makes two trips each of those days (round trip), is it better for them to buy the monthly pass or to pay for individual trips?

A group of 1st-year Bachillerato students is organizing a graduation party. They have calculated that the fixed costs of renting the venue and the DJ are $450$ €. In addition, the catering costs $12$ € per attendee. If they have raised a total of $1500$ € through raffle ticket sales and sponsorships, and they want to have at least $210$ € left over after the event for an end-of-year trip, write and solve an inequality that determines the maximum number of people, $x$, that they can invite.

In the graph you can see the money raised and the costs as a function of the number of attendees.

In a diving competition, the height $h$ (in metres) reached by an athlete with respect to time $t$ (in seconds) is given by the expression $h(t) = -t^2 + 4t + 1,5$. Determine the time interval during which the diver is at a height of more than $4,5$ metres.

Solve the following linear inequality, expressing the solution in interval notation and representing it on the number line:

A company that manufactures phone cases has determined that its monthly profits, in hundreds of euros, depend on the number of units produced $x$ (in thousands) according to the function $B(x) = -2x^2 + 14x - 12$.

Between which production values does the company make a profit (that is, $B(x) > 0$)? Solve the inequality and express the solution as an interval, in thousands of units.

Find the set of real numbers that simultaneously satisfy the following system of inequalities:

The area of a rectangular garden must be less than $54$ $m^2$. It is known that the length of the garden is $3$ metres longer than the width. If we call $x$ the measure of the width, set up a quadratic inequality and determine the possible values that this measure can take.

Solve the following rational inequality, taking special care with values that make the denominator zero:

Given the quadratic equation $x^2 - 6x + k = 0$, determine for which values of the real parameter $k$ the equation has no real solutions. Hint: Remember that for no real solutions to exist, the discriminant $\Delta = b^2 - 4ac$ must satisfy a specific inequality.

An economic consulting firm models the net return $R(x)$ of a small family business as a function of the number $x$ of product batches sold per month (with $x \ge 0$). After examining the financial statements, it is found that the company makes a strict profit (that is, $R(x) > 0$) precisely when the number of batches sold lies in the open interval $(3, 8)$. If production is exactly at $x = 3$ or at $x = 8$, the return is zero ($R(x) = 0$), and for any other sales level, losses occur. Furthermore, monthly fixed costs imply that if no batches are sold ($x = 0$), the losses amount to $48$ hundred euros (that is, $R(0) = -48$). Reasonably determine the quadratic inequality of the form $ax^2 + bx + c > 0$ that describes the company's profit zone. Clearly explain how you deduce the signs and the exact numerical values of the coefficients $a$, $b$, and $c$ from the roots and the constant term.

To solve the inequality $(2x - 5)^2 \le 9$, two Social Sciences students propose two different strategies:

• Marcos proposes expanding the square of the binomial, transposing $9$ to the left-hand side to obtain an expression of the form $ax^2 + bx + c \le 0$, applying the quadratic formula to find the two roots, and constructing a sign chart to deduce the solution interval.

• Sara proposes subtracting $9$ from both sides to write $(2x - 5)^2 - 3^2 \le 0$, applying the notable identity for the difference of squares $A^2 - B^2 = (A - B)(A + B)$ to factor the expression directly without using the general formula, or deducing that $|2x - 5| \le 3$.

Solve the inequality using the method proposed by Sara. Then, compare both approaches: justify which one is more direct in this case and discuss whether Sara's factoring approach would remain just as advantageous if the right-hand side had been a number that is not a perfect square, such as $7$, instead of $9$.

A consulting services firm estimates that its monthly profits, expressed in thousands of euros, are given by the rational expression $B(x) = \frac{x^2 - 9}{x - 5}$, where $x$ represents the number of projects managed simultaneously. For operational reasons, the firm can only manage between 0 and 10 projects. Determine the range of projects for which the firm achieves positive profits that are less than or equal to 4 thousand euros, that is, solve the system:

$0 < \frac{x^2 - 9}{x - 5} \leq 4$

In a sociological experiment on the spread of news on a social network, the number of people who know the news (in hundreds) after $t$ hours is modelled by the function $N(t) = -t^2 + 12t - 20$. At the same time, a moderation team intervenes by removing posts so that the maximum reach allowed by the platform's rules is given by $L(t) = 3t - 4$, that is, $N(t) < L(t)$ must hold.

Calculate the time interval (values of $t$ in hours) in which the news spreads naturally ($N(t) > 0$) but its reach already exceeds the limit allowed by the platform.

Determine for which values of the real parameter $k$ the quadratic inequality $x^2 + (k-2)x + k + 1 > 0$ holds for any real value of $x$. Apply this result to explain under what conditions an investment fund guaranteeing this minimum yield would never make a loss, regardless of market fluctuation $x$.

An electronic components manufacturer has found that the unit production cost $C$ depends on the room temperature $T$ (in ºC) according to the expression $C(T) = |T^2 - 40T + 375|$. For the product to be competitive in the market, the unit cost must not exceed 25 euros.

The figure shows the graph of $C(T)$ and the line $C = 25$.

Solve the inequality $|T^2 - 40T + 375| \leq 25$ and determine the temperature ranges allowed for manufacturing.

An NGO distributes food ($x$) and hygiene ($y$) packs. It has a maximum budget of 2000€. Each food pack costs 20€ and each hygiene pack costs 10€. For logistical reasons, they must send at least 40 food packs and the number of hygiene packs cannot exceed twice the number of food packs. Furthermore, for the aid to be balanced, the total sum of packs must be at least 100.

Write the system of linear inequalities in two variables that represents these constraints and graphically represent the feasible region in the plane.

You can do it on paper and send a photo with your answer.

Find the solution set of the following system of inequalities in one unknown, paying attention to simplifying fractions and studying the signs of the quadratic inequality:

$\begin{cases} \frac{x+3}{2} - \frac{x^2-1}{4} \geq 1 \\ |2x - 5| < 7 \end{cases}$

A rectangular garden is to be designed so that its perimeter is exactly 60 metres. If the area of the garden must be between 125 and 200 square metres, what values can the length of the longer side of the garden take? Express the result as an interval.

Solve the following quadratic rational inequality, making sure to correctly consider the domain of the expression and the intervals where the quotient changes sign:

$\frac{3}{x-2} \leq \frac{x}{x+3}$

State in detail the values that are excluded from the solution because they make the denominator zero.

A cooperative oil mill in Jaén packages extra virgin olive oil. The commercial team estimates that by setting a selling price of $p$ euros per jug (with $p > 20$), the monthly profit $B(p)$ (expressed in hundreds of euros) follows the parabolic model $B(p) = -2p^2 + 140p - 1600$. The members' assembly establishes as an essential target achieving a monthly profit of at least $600$ hundred euros ($60\,000$ €).

1. Estimate: Before performing formal algebraic calculations, find the coordinates of the vertex of the parabola to determine the price that maximizes profit and what this maximum profit is. Based on that information, argue without solving the inequality whether the cooperative's target is feasible and make a reasoned estimate of between which two approximate prices the jug should be priced.

2. Calculate: Set up the quadratic inequality that models the cooperative's requirement, solve it step by step by studying the sign on the real line, and write the exact interval of acceptable prices $[p_1, p_2]$.

3. Check and reflect: Check whether the interval obtained agrees with your previous estimate and explain, from an economic and social perspective, why prices higher than $p_2$ cause the profit to fall below the required threshold even though each jug is sold for more.

A socially responsible investment fund analyzes the feasibility of a portfolio whose temporal volatility index is modeled by the quadratic polynomial $V(t) = t^2 - 2(m + 1)t + (m^2 + 5m + 4)$, where $t$ represents time elapsed in quarters and $m$ is a regulatory adjustment parameter set by management. The risk committee requires volatility to be strictly positive at all times to prevent crash scenarios, that is, the quadratic inequality $t^2 - 2(m + 1)t + (m^2 + 5m + 4) > 0$ must hold for ANY real value of time ($t \in \mathbb{R}$).

Calculate with reasoning for which values of the parameter $m$ this requirement is guaranteed. Justify your answer by analyzing the geometric and analytical relationship between the sign of the leading coefficient, the condition on the discriminant $\Delta$ for there to be no real roots, and the position of the parabola with respect to the horizontal axis, explaining why it would not be mathematically valid to merely check a few random values of $t$.