Exercises: Linear inequalities in one variable

Solve the following linear inequality:

$3x - 5 < 2x + 4$

Find the solution set of the inequality by first removing the brackets:

$2(x - 3) + 4 \ge 5x - 1$

Solve the following inequality with denominators:

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

Translate into algebraic language and solve: "Twice a number minus three units is strictly less than the number itself plus five units."

Given the inequality $4x - 7 \ge 2x + 1$, check which of the following values belong to its solution set: $x = -2$, $x = 0$, $x = 4$, $x = 5$.

Aoife has solved the inequality $5 - 2x \le 11$ on the blackboard following these steps:

1) She subtracts $5$ from both sides: $-2x \le 6$.

2) She moves the $-2$ over by dividing: $x \le \frac{6}{-2}$.

3) She concludes that the solution is $x \le -3$, that is, the interval $(-\infty, -3]$.

However, when checking with the value $x = 0$ (which does not belong to her solution set), she observes that $5 - 2\times 0 = 5 \le 11$, which is undoubtedly a true inequality.

1. a) Identify and explain in a reasoned manner the conceptual error made by Aoife in her algebraic work.
2. b) Using the real number line and the reversal of order between opposite numbers, justify why the direction of the inequality must be reversed when dividing by a negative quantity.
3. c) Determine the correct solution set both as an inequality and in interval notation.

To solve the inequality $3 - 5x < 18$, two classmates propose two different strategies:

• Method A (Seán): He works by keeping the unknown on the left-hand side. He subtracts $3$ from both sides and then divides by $-5$, remembering to reverse the direction of the inequality to obtain $x > -3$.

• Method B (Aoife): She prefers to avoid dividing by negative numbers. She adds $5x$ to both sides to obtain $3 < 18 + 5x$ and solves for the unknown on the right-hand side without altering the direction of the inequality at any point.

1. a) Fully develop Aoife's procedure (Method B) and check that you reach exactly the same solution as with Method A.
2. b) Critically compare both methods: what advantage does Aoife's method offer to avoid typical sign errors, and what special precaution does it require when writing down the final solution interval from the obtained expression?

A small business manufactures mobile phone cases. The fixed monthly costs are $500$ euros and producing each case costs $2$ euros. If each case sells for $5$ euros, set up an inequality and determine the minimum number of cases that must be sold per month to make a profit of at least $1000$ euros.

The relationship between temperature in degrees Fahrenheit ($F$) and degrees Celsius ($C$) is given by the formula $F = \frac{9}{5}C + 32$. Set up an inequality and solve it to determine above what temperature in degrees Celsius $86$ degrees Fahrenheit is exceeded.

A Year 12 student has obtained the following marks in their first three mathematics tests: $6.5$, $7.0$ and $8.2$. What minimum mark must they get in the fourth test for their mean mark to be greater than or equal to $7.5$?

Consider the inequality $3x - a > 2x + 5$, where $a$ is a real parameter. Calculate the value of $a$ knowing that the solution set of the inequality is the interval $(8, \infty)$.

A car hire company (Co. A) charges a fixed fee of $30$ euros plus $0.15$ euros per kilometre travelled. A competing company (Co. B) charges a fixed $15$ euros plus $0.20$ euros per kilometre. Set up an inequality to find out from how many kilometres it is cheaper to hire the car from Co. A and interpret the result using the graph below.

A person living in Athlone regularly travels to Dublin for study purposes by Intercity train. A single one-way ticket costs $8.40\text{€}$. Irish Rail offers a monthly pass with a fixed subscription fee of $45\text{€}$, with which each individual ticket is reduced to a fare of $2.60\text{€}$.

1. a) Model the situation by setting up a linear inequality to determine the number of monthly trips, $x$, from which it is more economical to purchase the monthly pass than to buy individual single tickets.
2. b) Algebraically solve the inequality you set up.
3. c) Taking into account the nature of the variable (an integer number of trips), interpret the result and state precisely the minimum number of monthly trips that justify the economic benefit of buying the pass.

Given the inequality with denominators:

$\frac{2x - 1}{3} - \frac{x + 2}{4} \ge \frac{x}{2} - 1$

1. a) Before operating algebraically, make a reasoned estimate: evaluate the numerical value of the left-hand side and the right-hand side for the test integer values $x = 0$ and $x = 10$. From these two checks, deduce whether you expect the solutions of the inequality to correspond to small values (less than a bound) or to large values (greater than a bound).
2. b) Solve the inequality exactly by multiplying both sides by the least common multiple of the denominators, and express the resulting solution set in interval notation.
3. c) Check whether the result found is consistent with your initial estimate in part a), and explicitly verify whether the endpoint of the interval satisfies the corresponding equality.

Discuss and solve the inequality $kx - 3 \le 2x + k$ depending on the possible values of the real parameter $k$.

Solve the following literal inequality for the unknown $x$, assuming that the parameters satisfy the condition $a > b > 0$:

$\frac{x}{a} - \frac{x}{b} \ge \frac{a^2 - b^2}{ab}$

A telecommunications company offers two data plans. The Flat Rate plan costs $10$ euros per month and includes $2$ GB of data, charging $2$ euros for each additional GB. The Plus Plan costs $20$ euros per month and includes $5$ GB of data, charging $1.5$ euros for each additional GB. If a user knows that they always consume more than $5$ GB per month, what is the minimum number of gigabytes they must consume for the Plus Plan to be strictly cheaper than the Flat Rate plan?

A goods lift has a maximum capacity of $800$ kg. An operator weighing $75$ kg needs to transport boxes of building materials weighing $22$ kg each. If the operator must always accompany the load in the lift, set up and solve an inequality to determine the maximum number of whole boxes that can be transported in a single trip.

We want to determine the integer parameters $a$ and $b$ that define the linear inequality:

$\frac{ax + 5}{3} - b < x + 1$

knowing that two conditions are simultaneously satisfied:

1) Its solution set is exactly the open ray $(-\infty, 4)$.

2) If the left-hand side of the inequality is evaluated at $x = 1$, the expression $\frac{ax + 5}{3} - b$ takes exactly the value $1$.

1. a) Algebraically deduce the values of the coefficients $a$ and $b$ by setting up the necessary equations or relationships from the available information.
2. b) Explain what fundamental role the sign of the coefficient $(a - 3)$ plays when solving for $x$, and rigorously justify why, with the obtained values, the solution set is $(-\infty, 4)$ and not $(4, +\infty)$.

Consider the linear inequality in the unknown $x$ depending on the real parameter $m$:

$m(x - 2) + 3x \le 2m + 5(x - 1)$

1. a) Collect like terms and transform the inequality into the reduced form $(m - 2)x \le k(m)$, where $k(m)$ is a degree 1 polynomial in $m$.
2. b) Discuss in a reasoned manner the solution set of the inequality according to the different real values of $m$. Specify for which values of $m$ the solution is a ray towards $-\infty$, for which it is towards $+\infty$, and whether there is any value for which the inequality is satisfied by all real numbers ($S = \mathbb{R}$) or by none ($S = \emptyset$).
3. c) A high school student states: "Since for $m = 2$ the coefficient of $x$ becomes $0$, the inequality no longer makes sense and has no solutions at all." Critically analyse this statement by directly substituting $m = 2$ into the reduced expression and determine whether it is true or false.