Linear inequalities in one variable
Wilkin sells phone cases and wants to know how many he has to sell to make a profit. He is not looking for an exact number, but for "starting from how many." That is an inequality. It is solved almost like a first-degree equation, but with one extra rule: when you multiply or divide by a negative number, the inequality reverses direction.
The method: move terms and solve
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: $x = -2$, $x = 0$, $x = 4$, $x = 5$.
Solve the following linear inequality: $3x - 5 < 2x + 4$
Solve the following linear inequality:
$3x - 5 < 2x + 4$
Translate into algebraic language and solve: "Twice a number minus three units is strictly less than the number itself plus five units."
Translate into algebraic language and solve: "Twice a number minus three units is strictly less than the number itself plus five units."
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 does not 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: costs, temperature, grades, and rates
The most useful thing about inequalities is translating a statement ("at least," "must not exceed," "starting from how many") into an inequality.
| Phrase | Symbol |
|---|---|
| at least, a minimum of, no less than | $\ge$ |
| at most, a maximum of, no more than | $\le$ |
| more than, exceed | $>$ |
| less than | $<$ |
Steps: define the variable, set up the inequality, solve it, and interpret the answer (if it counts objects, it must be an integer).
The graph shows the idea: the line with the lower cost in each stretch is the better deal, and the intersection point marks the switch.
A small business manufactures cell phone cases. The fixed monthly costs are $30\,000$ pesos and producing each case costs $120$ pesos. If each case sells for $300$…
A small business manufactures cell phone cases. The fixed monthly costs are $30\,000$ pesos and producing each case costs $120$ pesos. If each case sells for $300$ 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 $60\,000$ 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…
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…
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 $1\,800$ pesos plus $9$ pesos per kilometer traveled. A competing company (Co. B) charges a fixed $900$ pesos plus…
A car rental company (Co. A) charges a fixed fee of $1\,800$ pesos plus $9$ pesos per kilometer traveled. A competing company (Co. B) charges a fixed $900$ pesos plus $12$ 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 $600$ pesos per month and includes $2$ GB of data, charging $120$ pesos for each additional…
A telecommunications company offers two data plans. The Flat Rate plan costs $600$ pesos per month and includes $2$ GB of data, charging $120$ pesos for each additional GB. The Plus Plan costs $1\,200$ pesos per month and includes $5$ GB of data, charging $90$ 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…
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.