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 $3\,000$ pesos, and the catering costs $240$ pesos per attendee. If they have a maximum budget of $12\,000$ pesos, 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 $200$ pesos plus $1$ peso per minute of call. Another company charges a fixed fee of $100$ pesos but $3$ pesos per minute. After how many minutes of calls per month does the first company become cheaper?

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 $450$ pesos. In addition, the catering costs $12$ pesos per attendee. If they have raised a total of $1500$ pesos through raffle ticket sales and sponsorships, and they want to have at least $210$ pesos 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 meters) 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 $4.5$ meters.

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

A cell phone case manufacturing company has determined that its monthly profits, in hundreds of pesos, depend on the number of units produced $x$ (in thousands) according to the function $B(x) = -40x^2 + 280x - 240$. Between which production values does the company make a profit (that is, $B(x) > 0$)?

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 $3$ meters 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.

A consulting services firm estimates that its monthly profits, expressed in thousands of pesos, 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 pesos, 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 pesos.

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 pesos. Each food pack costs 20 pesos and each hygiene pack costs 10 pesos. 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 meters. If the area of the garden must be between 125 and 200 square meters, 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.