Exercises: Inequalities

Write as an interval and represent on the number line the values of $x$ that satisfy the inequality $-3 \le x < 5$.

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

Solve the following linear inequality:

$4x - 5(x - 2) \ge 3x + 2$

Solve the quadratic inequality $x^2 - 16 < 0$. Express the solution as an interval.

Solve the following system of inequalities in one variable:

$\begin{cases} 2x - 3 > 1 \\ x + 4 \le 7 \end{cases}$

Translate into algebraic language and solve: "Three times a number, decreased by $4$ units, is at most $11$".

Check whether the points $A(2, -1)$, $B(0, 3)$ and $C(-1, -2)$ are solutions to the inequality in two variables $3x - 2y \le 4$.

Solve the absolute value inequality $|x - 3| \le 5$ and express its solution using an interval.

Solve the linear inequality with denominators:

$\frac{x}{2} - \frac{x - 1}{3} > 1$

Marcos has to solve the linear inequality $-3x + 12 \ge 0$ and writes the following steps in his notebook:

$-3x + 12 \ge 0$

$-3x \ge -12$

$x \ge \frac{-12}{-3}$

$x \ge 4$

He concludes by stating that the solution set is the interval $[4, +\infty)$.

1. a) Without solving the inequality yet, check what happens if you substitute a value from Marcos's set into the original inequality, for example $x = 5$. Is the inequality satisfied?
2. b) Identify and explain with reasoning which rule of inequalities Marcos forgot to apply in the step from the second to the third line. Why does the direction of an inequality reverse when multiplying or dividing by a negative number?
3. c) Correct the procedure and write the correct solution set in interval notation.

The quality control department of an olive oil bottling plant in Jaén programs its machinery to fill $1$-liter bottles. A bottle is accepted if the actual volume of oil, $x$ (in liters), lies within the interval $[0,98;\ 1,02]$.

1. a) Express this acceptance condition using an absolute value inequality of the form $|x - a| \le k$, deducing and indicating with reasoning which number represents the center $a$ and what the tolerance margin $k$ is.
2. b) Conversely, a bottle is considered defective and is rejected if it does not meet this standard. Write the absolute value inequality that describes the rejected bottles and express the set of defective volumes as the union of two disjoint intervals.

Solve the linear inequality with fractions and brackets:

$\frac{3(x - 2)}{4} - \frac{2x + 1}{6} \le \frac{x - 3}{2} + 1$

Solve the quadratic inequality $-2x^2 + 5x + 3 \ge 0$. Express the solution as an interval.

Find the solution to the following system of inequalities:

$\begin{cases} \frac{x + 1}{2} - \frac{x - 2}{3} > x - 1 \\ 3x - 4(x + 1) \ge -6 \end{cases}$

Write the system of inequalities in two variables whose solution corresponds to the shaded region in the following figure.

A farmer wants to bottle his oil production. Knowing that he has $2$-litre bottles and $5$-litre bottles available, and that he has strictly less than $100$ litres of oil, write a linear inequality in two variables that models this situation. If he has already filled $10$ $5$-litre bottles, what is the maximum number of $2$-litre bottles he can fill?

The base of a rectangle measures $5$ cm more than its height. If the perimeter of the rectangle must be greater than $30$ cm but its area cannot exceed $150$ cm$^2$, between what values does the measure of the height lie?

Solve the following rational inequality, analyzing the sign in the corresponding intervals:

$\frac{2x - 5}{x + 3} \le 0$

To pass a subject, a student must have a mean mark of at least $5$ marks across the three term exams. If in the first two exams they scored $3,5$ and $4,2$, what mark must they get in the third exam to pass? (The maximum mark is $10$).

To travel to her high school and extracurricular activities during the month of October, Lucía is considering two public transit options:

• Option A (Reloadable card): pays €$2,50$ for the physical card (one-time payment) and €$1,35$ for each individual trip taken.

• Option B (Monthly youth pass): flat rate of €$21,40$ per month with unlimited trips.

1. a) Estimate: without setting up equations, mentally calculate the cost of 10 and 20 trips with Option A and estimate between which tens of trips the youth pass will start to be more cost-effective. Explain your estimate.
2. b) Calculate: set up an inequality where the unknown $x$ is the integer number of trips needed for Option B to be strictly cheaper than Option A. Solve it step by step.
3. c) Verify: determine the exact minimum number of trips at which it is worth buying the pass and compare this result with your previous estimate. If Lucía plans to take only 12 trips in the entire month, which option would you advise her to choose?

The Directorate-General for Traffic (DGT) warns that on wet asphalt, braking distance increases significantly. For a passenger car, the braking distance $d$ (in meters) as a function of driving speed $v$ (in km/h) can be modeled using the approximate formula:

$d = 0,005 \cdot v^2 + 0,2 \cdot v$

In an urban school zone in the rain, the maximum distance available to brake in an emergency without overshooting the pedestrian crossing is $30$ meters (that is, $d \le 30$).

1. a) Model the situation by setting up a quadratic inequality with respect to speed $v$.
2. b) Transform the inequality to have a zero on the right-hand side, find the roots of the associated quadratic equation, and construct the corresponding sign chart to determine its mathematical solution set.
3. c) Interpret the resulting interval considering the physical reality of the problem: what is the range of safe speeds at which the vehicle can travel? Justify why the mathematical interval must be bounded on the left.

Determine the values of the parameter $m$ so that the quadratic inequality $x^2 - mx + 9 > 0$ holds for any real number $x$.

Solve the higher-degree polynomial inequality by first factorising the polynomial:

$x^3 - 4x^2 - x + 4 \le 0$

Solve the rational inequality, paying attention to simplifying correctly before studying the signs (remember that you cannot eliminate denominators containing the unknown):

$\frac{x + 1}{x - 2} > \frac{1}{2}$

A small company manufactures two types of wireless headphones: model Alfa and model Beta. Model Alfa requires $3$ hours of assembly and $1$ hour of calibration, while model Beta requires $2$ hours of assembly and $2$ hours of calibration. The company has a maximum of $120$ hours for assembly and $80$ hours for calibration per week. Set up the system of inequalities that defines the feasible region and state whether it would be possible to manufacture $20$ Alfa headphones and $25$ Beta headphones in a week.

Observe the following bounded feasible region (a triangle) and determine the system of three linear inequalities in two unknowns that defines it.

A toy rocket is launched vertically upwards. The height $h$ (in metres) reached by the rocket as a function of time $t$ (in seconds) is given by the expression $h(t) = 30t - 5t^2$. During which time interval will the rocket be at a height strictly greater than $25$ metres?

Solve the following inequality combining modulus and quadratic polynomials:

$|x^2 - 5x| < 6$

A balanced diet requires that the daily intake of two foods, $A$ and $B$, provides at least $60$ g of protein and does not exceed $80$ g of fat. Each $100$ g portion of food $A$ contains $15$ g of protein and $10$ g of fat. Each $100$ g portion of food $B$ contains $10$ g of protein and $20$ g of fat. If we let $x$ and $y$ be the number of $100$ g portions of food $A$ and $B$ consumed respectively, set up the system of inequalities and find out whether it is possible to consume exactly $3$ portions of $A$ and $2$ portions of $B$ while satisfying the constraints.

To solve the quadratic inequality $x^2 - 2x - 8 \le 0$, two classmates propose different strategies:

• Method A (Algebraic): find the roots of $x^2 - 2x - 8 = 0$, factor as $(x - 4)(x + 2) \le 0$, divide the real number line into three intervals, and create a sign chart by evaluating test points.

• Method B (Geometric / Functional): interpret the expression as the parabolic function $y = x^2 - 2x - 8$, observe that its branches open upwards ($a = 1 > 0$) and that it intersects the $X$-axis at its roots, visually deducing the section where the curve lies below or on the axis ($y \le 0$).

1. a) Solve the inequality using both methods and verify that the same solution set is obtained.
2. b) Justify which of the two approaches you consider more efficient and less prone to calculation errors with signs on an exam.
3. c) Now consider the inequality $x^2 - 2x + 5 \le 0$. When calculating its roots, it turns out that the discriminant is negative (there are no real roots). Explain how Method B allows you to deduce the solution almost immediately without resorting to sign charts, and determine the solution to this inequality with reasoning.

Given the following system of linear inequalities in one variable involving a real parameter $k$:

$\begin{cases} 2(x - 1) < x + 3 \\ 3x + k \ge 7 \end{cases}$

1. a) Solve the first inequality separately and express its solution as an open interval.
2. b) Solve for the unknown $x$ in the second inequality in terms of the parameter $k$. Then, solve the inverse problem: determine what exact value $k$ must take so that the final solution of the system (the intersection of both inequalities) is exactly the half-open interval $[-1, 5)$.
3. c) Analyze what happens if the value of $k$ varies: for what values of $k$ is the system inconsistent (has no solution)? Justify your conclusion using the graphical representation of intervals on the real number line and the definition of the intersection of sets.