Linear Inequalities in One Variable
Ximena wants to rent a car for a trip from CDMX to Puebla, and two companies offer her different rates. After how many kilometers is one a better deal than the other? A linear inequality is solved almost like a first-degree equation, with one extra rule that changes everything: when you multiply or divide by a negative number, the inequality reverses direction.
The method: move terms and solve
The solution is written as an interval: an open circle or parenthesis when the endpoint is not included ($<$, $>$), a closed circle or bracket when it is ($\le$, $\ge$).
Solve the following linear inequality: $3x - 5 < 2x + 4$
Solve the following linear inequality:
$3x - 5 < 2x + 4$
Given the inequality $4x - 7 \ge 2x + 1$, check which of the following values belong to its solution set: $x = -2$, $x = 0$, $x = 4$, $x = 5$.
Given the inequality $4x - 7 \ge 2x + 1$, check which of the following values belong to its solution set: $x = -2$, $x = 0$, $x = 4$, $x = 5$.
With parentheses and denominators
If there are parentheses, expand them first. If there are denominators, multiply the whole inequality by the LCM, which is positive, so the direction doesn't change in that step.
Find the solution set of the inequality by first removing the brackets: $2(x - 3) + 4 \ge 5x - 1$
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: $\frac{x}{2} - 1 \le \frac{3x}{4} + 2$
Solve the following inequality with denominators:
$\frac{x}{2} - 1 \le \frac{3x}{4} + 2$
Problems: translating into algebra
The most useful thing about inequalities is translating a word problem into an inequality and solving it. These phrases come up again and again:
| Phrase | Symbol |
|---|---|
| "at least," "a minimum of," "equal to or greater than" | $\ge$ |
| "at most," "a maximum of," "must not exceed" | $\le$ |
| "more than," "greater than," "exceeds" | $>$ |
| "less than," "strictly less than" | $<$ |
For example, "twice a number minus three is strictly less than the number plus five" is written $2x-3 Let's go back to Ximena. The two companies charge the following (with $x$ in kilometers): Translate into algebraic language and solve: "Twice a number minus three units is strictly less than the number itself plus five units." 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 $7.5$? 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. 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.Translate into algebraic language and solve: "Twice a number minus three units is strictly less than the number itself plus five units."
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,…
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…
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…
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 $4$…
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…
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…
Inequalities with parameters
When another letter besides $x$ appears (a parameter), you have to look at the sign of the coefficient of $x$ before dividing. If it can be positive, negative, or zero, the solution depends on the case.
Another typical situation is the reverse: you're given the solution and must find the parameter. In $3x-a>2x+5$ we get $x>5+a$. If the solution is $(8,\infty)$, then $5+a=8$, so $a=3$.
If the parameter appears in the denominators and its sign is known (for example $a>b>0$), that information tells you whether the direction changes or not when you multiply or divide. Before anything else, factor or gather the terms with $x$.
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…
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 $(8, \infty)$.
Discuss and solve the inequality $kx - 3 \le 2x + k$ depending on the possible values of the real parameter $k$.
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 $a > b > 0$:…
Solve the following literal inequality for the unknown $x$, assuming that the parameters satisfy the condition $a > b > 0$:
$\frac{x}{a} - \frac{x}{b} \ge \frac{a^2 - b^2}{ab}$
Back to the beginning: Ximena should choose Company A if she plans to drive more than 300 km, and Company B if she'll drive less.