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: $x = -2$, $x = 0$, $x = 4$, $x = 5$.

A small business manufactures cell phone cases. The fixed monthly costs are $10\,000$ pesos and producing each case costs $40$ pesos. If each case sells for $100$ pesos, set up an inequality and determine the minimum number of cases that must be sold per month to make a profit of at least $20\,000$ pesos.

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 rental company (Co. A) charges a fixed fee of $600$ pesos plus $3$ pesos per kilometer traveled. A competing company (Co. B) charges a fixed $300$ pesos plus $4$ pesos per kilometer. Set up an inequality to find out from how many kilometers it is cheaper to rent the car from Co. A and interpret the result using the graph below.

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 $200$ pesos per month and includes $2$ GB of data, charging $40$ pesos for each additional GB. The Plus Plan costs $400$ pesos per month and includes $5$ GB of data, charging $30$ pesos 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.