Linear inequalities in one unknown
A linear inequality is solved in the same way as a first-degree equation, with one extra rule that changes almost everything: when you multiply or divide by a negative number, the inequality reverses its direction. Mastering this thoroughly is the basis of everything that comes later in the topic.
The method: move terms across and solve for x
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 brackets and denominators
If there are brackets, expand them first. If there are denominators, multiply the whole inequality by the LCM, which is always 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: tariffs, marks and translating into algebraic language
The most practical use of linear inequalities is to translate a statement ("at least", "must not exceed", "from how many onwards") into an inequality and solve it.
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…
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.
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 $7,5$?
Inequalities with parameters
When the inequality has a letter besides $x$ (a parameter), you have to discuss the sign of what multiplies $x$ before solving for it: if that coefficient can be positive or negative, the solution depends on the case.
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 $k$.