Exercises: Linear inequalities with one unknown

Solve the following linear inequality:

$3x - 5 < 2x + 4$

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

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

Solve the following inequality with denominators:

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

Find the values of $x$ that satisfy the inequality:

$4 - 3x > 10$

Determine the smallest integer $x$ that satisfies the following inequality:

$5x + 7 > 22$

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

Simplify and solve the linear inequality:

$x + 2x + 3x < 12 + 2x$

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 company manufactures cell phone cases. The fixed monthly costs are $\$500$ and producing each case costs $\$2$. If each case is sold for $\$5$, set up an inequality and determine the minimum number of cases that must be sold per month to obtain a profit of at least $\$1000$.

Solve the following inequality combining the use of parentheses and denominators:

$\frac{2(x-1)}{3} - \frac{3x+1}{4} \le \frac{5-x}{6}$

The relationship between the 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 from what temperature in degrees Celsius $86$ degrees Fahrenheit is exceeded.

A high school student has obtained the following grades on their first three math exams: $6.5$, $7.0$, and $8.2$. What minimum grade must they get on the fourth exam for their average grade 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 $(8, \infty)$.

Solve the following inequality by first working with decimal numbers:

$0.5(x - 2) + 0.25x \ge 1.5x - 0.75$

In a rectangle, it is known that the base measures $5$ in. more than the height. If the perimeter of the rectangle must be strictly less than $50$ in., what are the possible values that the height measurement can take?

A car rental company (Co. A) charges a flat fee of $\$30$ plus $\$0.15$ per mile traveled. Another competing company (Co. B) charges a flat $\$15$ plus $\$0.20$ per mile. Set up an inequality to find out at how many miles it becomes cheaper to rent the car from Co. A and interpret the result using the following graph.

Analyze and solve the inequality $kx - 3 \le 2x + k$ in terms of the possible values of the real parameter $k$.

A laboratory has $10$ liters of an acid solution at $20\%$. Set up an inequality to calculate how many liters of pure acid (at $100\%$) must be added so that the concentration of the new mixture is at least $50\%$.

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

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

An investor has $\$20000$ in capital. They decide to invest part in a low-risk fund that offers a $4\%$ annual return, and the rest in a technology fund that offers a $6\%$ annual return. What is the minimum amount they must invest in the technology fund to ensure that the total interest generated after one year is at least $\$1000$?

Determine the value of the parameter $m$ so that the solution set of the inequality

$\frac{mx - 1}{2} - \frac{x + m}{3} > 1$

is exactly the interval $(4, \infty)$.

A telecommunications company offers two data plans. The Standard Plan costs $\$10$ per month and includes $2$ GB of data, charging $\$2$ for each additional GB. The Plus Plan costs $\$20$ per month and includes $5$ GB of data, charging $\$1.5$ for each additional GB. If a user knows 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 Standard Plan?

A freight elevator has a maximum capacity of $800$ pounds. A worker weighing $75$ pounds must take up boxes of building materials weighing $22$ pounds each. If the worker must always accompany the load in the elevator, set up and solve an inequality to determine the maximum number of whole boxes they can transport in a single trip.

Solve the following inequality and justify the result obtained analytically:

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