Exercises: Linear inequalities

Solve the inequality $3x - 5 < 7$ and express its solution set in interval notation.

Solve the inequality $-2x + 4 \geq 10$ and state at which step of the solution the direction of the inequality sign flips and why this happens.

The figure shows the solution set of an inequality on the number line.

1. a) Express it as an interval. b) Express it using an inequality. c) Write a linear inequality whose solution is exactly that set.

A car park charges $2\ \text{€}$ for entry plus $1{,}20\ \text{€}$ per hour. a) Write the inequality that allows you to calculate the number of hours $t$ you can park without exceeding $15\ \text{€}$. b) Solve it and interpret the result.

Solve the inequality $5(x - 2) \leq 3x + 4$.

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

A classmate claims: "to solve $-3x > 6$, I divide both sides by $-3$ and get $x > -2$". Decide whether their statement is true or false, explain the error if there is one, and give the correct solution.

Check whether the values $x = 2$ and $x = 8$ are solutions to the inequality $5(x - 1) - 2x < 10$, without solving it beforehand. Then, solve it and confirm your answer.

Lucía solved the inequality $5 - 2x \leq 11$ on the board following these steps:

1) Transposes the $5$: $-2x \leq 11 - 5 \implies -2x \leq 6$.

2) Solves for $x$: $x \leq \frac{6}{-2} \implies x \leq -3$, concluding that the solution is $(-\infty, -3]$.

1. a) Identify the conceptual error Lucía made in step 2 and explain why this happens using a simple numerical counterexample (for example, check what happens to the true inequality $2 < 6$ if you divide both sides by $-2$).
2. b) Correct the procedure, write the correct solution as an interval, and justify whether the endpoint $-3$ should be included or not.

In a public transport company in a Spanish city, a single bus ticket costs $1,40$ €. There is also a monthly travel card with a flat rate for unlimited trips for $21$ €, although to activate it for the first time you must pay a non-refundable initial deposit of $3,50$ € for the card itself.

1. a) Set up a linear inequality that represents from how many trips $x$ during that first month it becomes more economical to purchase the monthly pass than to buy single tickets.
2. b) Solve the inequality, express the solution as an interval, and determine the minimum integer number of trips needed to break even on the travel pass in the first month.

Solve the inequality $\frac{2x - 1}{3} - \frac{x + 2}{4} \leq \frac{1}{2}$ and state the largest integer that satisfies it.

Solve the double inequality $-3 \leq 2x + 1 < 7$ and express the solution as an interval.

Solve the system of inequalities $\begin{cases} 2x - 3 < 5 \\ x + 4 \geq 1 \end{cases}$ and represent the solution on the number line.

The figure shows two intervals $A$ and $B$ represented on the number line.

Write, as an interval, the set of values of $x$ that belong to both $A$ and $B$ at the same time, that is, the solution to the system formed by the two corresponding inequalities.

Ainhoa wants to get a mean score of at least $6{,}5$ across the three mathematics exams this term. In the first two, she scored $5{,}6$ and $7{,}2$. What minimum mark does she need in the third exam?

Two phone companies offer the following tariffs: Company A charges a monthly fee of $15\ \text{€}$ plus $0{,}03\ \text{€}$ per minute, and Company B charges no fee but $0{,}08\ \text{€}$ per minute. a) Write the monthly cost for each company as a function of the minutes $m$ used. b) For what monthly usage do both cost the same? c) For what values of $m$ is Company A cheaper?

A student solves the inequality $\frac{x - 3}{2} - \frac{2x + 1}{6} \geq 1$ as follows:

"I multiply everything by $6$: $3(x - 3) - (2x + 1) \geq 6$; I expand the brackets: $3x - 9 - 2x + 1 \geq 6$; I simplify: $x - 8 \geq 6$; solution: $x \geq 14$."

Identify the error made, explain it, and find the correct solution.

A rectangular garden is $7$ m wide. a) If less than $40$ m of fencing is available to enclose its perimeter, what values can the length $l$ take? b) If we also want the area of the garden to exceed $70\ \text{m}^2$, between which values must the length lie?

You are going to design an exam exercise for your classmates on inequalities:

1. a) Construct a linear inequality of the form $ax + b > c$, where $a$, $b$, and $c$ are integers, with the essential condition that the coefficient $a$ is negative ($a < 0$), such that its solution set is exactly the interval $(-\infty, 4)$. Explain with reasoning how you chose the values of $a$, $b$, and $c$ so that the direction of the inequality is reversed when solving for the variable.
2. b) Add a very simple second linear inequality to yours, such that together they form a system whose solution is the bounded interval $(-1, 4)$. Justify your answer by sketching or representing the intersection of both intervals.

The State Meteorological Agency (AEMET) establishes for a mountain area that a yellow warning for low temperatures is activated when the temperature drops below $-6$ ºC, and turns into an orange warning if it drops below $-12$ ºC. At 19:00, a mountain shelter records $3$ ºC, but the arrival of a cold front leads to the forecast that the temperature will drop at a constant rate of $1,5$ ºC for each elapsed hour ($t$).

1. a) Set up a compound inequality that determines during which hours the area will be at yellow warning level (that is, with a temperature strictly lower than $-6$ ºC, but not lower than $-12$ ºC, i.e., greater than or equal to $-12$ ºC).
2. b) Solve the compound inequality by operating simultaneously on all three parts, write the result as an interval, and interpret between what exact hours of the early morning the yellow warning will be in effect.

A student faces the modulus inequality $|3x - 2| < -4$. To solve it, they apply a mechanical rule by splitting the problem into two cases: $3x - 2 < -4$ and $3x - 2 > 4$. After solving both linear inequalities, they conclude that the solution is the set $x \in (-\infty, -2/3) \cup (2, +\infty)$. Explain the fundamental theoretical error they made from the very first step by ignoring the basic definition of absolute value (modulus) and provide the correct reasoned solution.

Find all values of the parameter $k$ for which $x = 3$ is a solution to the inequality $kx - 2(x - k) > 4$.

1. a) Solve the inequality $2(x - 1) > 2x + 5$. b) Solve the inequality $3x - 3 \leq 3(x + 5)$. c) In light of the previous parts, explain which two special cases can arise when an inequality reduces to an expression of the form $0 \cdot x > a$ or $0 \cdot x \leq a$, and what each depends on.

The figure shows the solution set of an inequality of the form $|x - a| \leq b$ on the number line.

1. a) Determine the values of $a$ and $b$. b) Solve the inequality $|2x - 3| \leq 5$ by transforming it into a double inequality.

To move the school's equipment, two van hire companies are compared: KmCero charges $45\ \text{€}$ per day plus $0{,}25\ \text{€}$ per kilometre, and RutaPlana charges $70\ \text{€}$ per day plus $0{,}10\ \text{€}$ per kilometre. a) Write the cost for one day's hire for each company as a function of the distance in kilometres $x$ driven. b) If it is expected to drive $120$ km in one day, which company is better value? c) Above how many kilometres per day is RutaPlana cheaper? d) A teacher claims: "we always win with KmCero because its fixed price is lower". Explain, using your results, why they are wrong.

A music streaming platform has fixed costs of $18.000\ \text{€}$ per month plus $1{,}20\ \text{€}$ in technical costs per subscriber. Each subscriber pays a monthly fee of $4{,}80\ \text{€}$ and advertising brings in $6.000\ \text{€}$ per month. a) Set up the inequality that determines the number of subscribers $x$ above which the platform makes a profit. b) Solve it and interpret the result.

The solution to the inequality $ax + b > 0$ is exactly the interval $(-\infty, 2)$, and it is known that $a + b = 4$. a) Reason why the coefficient $a$ must be negative. b) Find the values of $a$ and $b$.

To pass an online course, a student must achieve a mean mark between $7$ and $8{,}5$ across the three final tests. Marc obtained $6{,}4$ and $8{,}1$ in the first two tests. a) Set up the inequalities that his score $x$ in the third test must satisfy. b) Between which values must $x$ lie? c) Given that the maximum score for each test is $10$, interpret the result: can Marc pass the course? With what marks would he achieve it?

Let $a$ and $b$ be two negative numbers such that $a < b < 0$. a) Choose two specific values for $a$ and $b$ and calculate $a^2$, $b^2$ and $a \cdot b$. b) Conjecture an ordering from smallest to largest of the three numbers $a^2$, $b^2$ and $a \cdot b$. c) Prove your conjecture using the properties of inequalities.

A 4th-year ESO class is organising an end-of-year trip. A coach company offers them a quote with a fixed cost of $650$ € plus $0,80$ € for each kilometre travelled. The maximum budget the group has for transport is $1100$ €. The fixed total round-trip distance to the destination is $420$ km, and the rest of the budget will be used for excursions around the area during their stay.

1. a) Estimate: Before solving anything algebraically, round the data mentally to roughly estimate how many additional kilometres they will be able to spend on excursions without exceeding the available money. Write down your reasoning and your estimated value.
2. b) Model and check: Let $x$ be the additional kilometres of excursions. Set up an inequality to ensure they do not exceed the budget of $1100$ €, solve it step by step, and express the result as an interval. Compare the exact solution with your previous estimate and critically assess whether your initial forecast was accurate.

To solve the inequality $\frac{2 - 3x}{6} - \frac{x + 1}{3} \geq \frac{1}{2}$, two students suggest different paths:

• Marcos's strategy: Multiplies all terms by the LCM ($6$) to eliminate denominators from the start, writing $(2 - 3x) - 2(x + 1) \geq 3$.

• Sara's strategy: Moves all terms with $x$ to the right-hand side to ensure the coefficient of the unknown remains positive and avoid dividing by negatives: $\frac{2}{6} - \frac{1}{3} - \frac{1}{2} \geq \frac{3x}{6} + \frac{x}{3}$.

1. a) Solve the inequality by developing both methods completely and verify that exactly the same solution interval is obtained.
2. b) Analyse the advantages and disadvantages of each method regarding the two most common mistakes in this type of exercise: forgetting to change signs when eliminating a fraction preceded by a minus sign, and forgetting to reverse the inequality when dividing by a negative number. Justify which of the two strategies you would recommend in order to make fewer mistakes on an exam.