Exercises: Inequalities with one unknown

Solve the following linear inequality: $3(x - 2) + 4 \leq 5x - 8$. Express the solution in interval notation.

Solve the inequality with denominators: $\frac{x}{2} - \frac{x-1}{3} > 1$.

Find the values of $x$ that satisfy the quadratic inequality, which is already given in factorised form: $(x - 4)(x + 2) < 0$.

Solve the following incomplete quadratic inequality and represent its solution on the number line: $2x^2 - 18 \geq 0$.

Find the solution interval of the quadratic inequality: $x^2 - 5x + 6 \leq 0$.

Solve the following system of inequalities with one unknown and express the solution as a single interval:

$2x - 3 < 5$

$3x + 1 \geq -2$

Solve the simple rational inequality by studying the signs of the numerator and denominator: $\frac{x-3}{x+1} \leq 0$.

The sum of double a natural number and $5$ is strictly less than $17$. Set up the corresponding inequality and find all the natural numbers that satisfy it.

Marcos is solving the rational inequality $\frac{6}{x - 2} \leq 2$. To eliminate the denominator, he decides to multiply both sides by $(x - 2)$, obtaining $6 \leq 2(x - 2)$. After operating, he concludes that the solution is $x \geq 5$, that is, the interval $[5, +\infty)$. a) Identify the conceptual error Marcos makes in his first step and explain why it is not valid to multiply both sides of an inequality by an algebraic expression without knowing its sign. b) Find a numerical counterexample (a value of $x$) that satisfies the original inequality but does not belong to Marcos's solution. c) Solve the inequality correctly by moving all terms to the left-hand side, combining them into a single fraction, and studying the resulting signs.

Consider the quadratic polynomial inequality $x^2 - 2x - 8 < 0$. a) Before carrying out any algebraic calculation, estimate the numerical sign of the polynomial by mentally evaluating simple values such as $x = -3$, $x = 0$, and $x = 5$. With this estimation, predict with justification whether the solution set will be a bounded interval of the form $(a, b)$ or the union of two infinite half-lines. b) Solve the inequality step by step using the factorisation of the polynomial and a sign chart. c) Check whether the interval found matches your initial prediction and justify whether the boundary values $x = -2$ and $x = 4$ are part of the solution set or not.

Solve the following inequality, paying attention to the negative signs in front of the fractions: $\frac{2x - 1}{4} - \frac{3x + 2}{6} \geq \frac{x - 5}{3} - 1$.

A rectangular garden is to be built such that its length is $4$ metres greater than its width $x$. If there is enough fertiliser to cover a maximum of $45 \text{ m}^2$, set up an inequality and determine the possible values that the width $x$ can take.

A small craft business has monthly fixed costs of $300$ euros and producing each leather bracelet costs $2$ euros. If it sells each bracelet for $5$ euros, how many bracelets must it sell per month as a minimum to start making a profit? Set up and solve the inequality.

Solve the rational inequality: $\frac{2x + 1}{x - 2} > 3$. Remember to move all terms to one side and simplify to obtain a single fraction before studying the signs.

Solve the higher-degree polynomial inequality by first taking out a common factor: $x^3 - 4x^2 - 5x \leq 0$.

Solve the inequality with modulus and express its solution as a union of intervals: $|2x - 5| \geq 7$.

A delivery van consumes $8$ litres of fuel every $100 \text{ km}$ on the highway and $12$ litres every $100 \text{ km}$ in the city. If on a journey of $500 \text{ km}$ in total it has consumed less than $45$ litres, what is the maximum number of kilometres it could have driven in the city?

Determine for which values of the parameter $k$ the quadratic equation $x^2 - kx + 9 = 0$ has no real solutions. Set up the corresponding inequality using the discriminant and solve it.

A university student from Valencia needs to commute daily to her faculty using Metrovalencia. For the zone she travels through, she is presented with two fare options: Option A, buying single tickets at $2,80$ € per trip; or Option B, purchasing a monthly pass for unlimited travel for $36$ € per month, to which she must add a one-off payment of $4$ € in the first month for the physical issuance of the personalised card. a) Set up a linear inequality with one unknown that determines the number of trips per month from which Option B becomes strictly cheaper than Option A during the first month of use. b) Solve the inequality and interpret the solution by indicating the minimum integer number of trips required to make the pass worthwhile. c) If the teaching month has 21 days and she makes 2 trips daily (round trip), calculate how much money she saves monthly with the more favourable option compared to the other.

To solve the absolute value inequality $|2x - 6| \leq 10$, two different strategies are proposed: Method 1 (Algebraic method), transform the expression into the simultaneous compound inequality $-10 \leq 2x - 6 \leq 10$ and solve for $x$; and Method 2 (Geometric interpretation), factor out the common factor to write $2 \cdot |x - 3| \leq 10$, simplify to $|x - 3| \leq 5$, and interpret the distance to the central value $3$ on the real number line. a) Work out both procedures explicitly and verify that the exact same solution interval is obtained. b) Reasonably justify which of the two methods you consider more efficient and which one minimizes the likelihood of making sign errors. c) Explain how the setup of each method would be adapted if the inequality were strict and had the opposite direction: $|2x - 6| > 10$.

Solve the rational inequality where the numerator and the denominator are quadratic polynomials: $\frac{x^2 - 2x - 8}{x^2 - 9} \leq 0$.

Solve the following system of inequalities and explain the special feature of its solution set:

$x^2 - 4x + 4 \leq 0$

$2x - 1 > 0$

A water rocket is launched vertically upwards in a school experiment. Its height $h$ in metres as a function of time $t$ in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine algebraically during which time interval the rocket will be above a height of $17 \text{ m}$.

Solve the modulus inequality that includes fractional expressions: $\left| \frac{3x - 1}{2} \right| < 4$.

The monthly profit of an electric scooter factory, expressed in thousands of euros, is given by the function $B(x) = -x^3 + 10x^2 - 16x$, where $x$ represents the number of hundreds of scooters manufactured. How many scooters must be manufactured so that the company does not make a loss (that is, so that $B(x) \geq 0$)?

In an isosceles triangle, the equal sides measure $x \text{ cm}$ and the third side measures $10 \text{ cm}$. For the triangle to be constructed, the sum of the lengths of any two sides must be strictly greater than the length of the third side. Furthermore, its perimetre must not exceed $36 \text{ cm}$. Set up a system of inequalities and find the possible values of $x$.

Study and solve the following system consisting of a linear inequality and a quadratic inequality:

$3x - 5 \leq x + 1$

$x^2 - 7x + 10 > 0$

Solve the following rational inequality, simplifying the algebraic additions required to express it as a single quotient before studying the signs: $\frac{1}{x} + \frac{1}{x+2} \geq \frac{3}{4}$.

The aim is to design a rational inequality in one unknown whose exact solution set is $(-3, 2] \cup (5, +\infty)$. a) Construct an inequality of the form $\frac{P(x)}{Q(x)} \geq 0$, where both the numerator $P(x)$ and the denominator $Q(x)$ are polynomials factored into monic linear binomials. Justify step by step, relying on the study of signs by intervals, the position of each factor in the numerator or in the denominator. b) Explain in detail why the numbers $-3$ and $5$ must appear with an open interval (parenthesis) in the solution, while $2$ is closed (with a bracket). c) Without redoing the entire sign chart, reason what elementary transformation you should apply to your inequality so that the solution becomes $(-\infty, -3) \cup [2, 5)$.

An automotive component manufacturer in Navarra designs a thermal sensor whose operating electrical resistance $R$ (in ohms) depends on the outdoor temperature $t$ (in degrees Celsius) according to the expression $R(t) = t^2 - 2(k - 1)t + (k + 5)$, where $k$ is a real calibration parameter. Due to strict safety standards, the device only operates correctly if the resistance is strictly positive ($R(t) > 0$) for any temperature $t \in \mathbb{R}$. a) Interpret the graph of $y = R(t)$ as a parabola and deduce what geometric condition with respect to the x-axis and what algebraic condition on its discriminant $\Delta = b^2 - 4ac$ must be satisfied to ensure such operability. b) Set up and solve the quadratic inequality in the unknown $k$ resulting from the previous condition, indicating the interval of valid values for the calibration of the sensor. c) If a factory error occurs and the parameter is set to $k = 4$, justify what anomaly will occur and determine at what exact temperature $t$ the sensor will stop working due to its resistance vanishing.