Linear Inequalities
Class 11 Mathematics · Chapter 5
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Summary
Many real situations are described not by equations but by inequalities using the symbols \(<,\ >,\ \le,\ \ge\). This chapter studies linear inequalities in one variable. A solution is any value of the variable that makes the inequality true, and the collection of all such values is the solution set, which may be found over \(\mathbb{N}\), \(\mathbb{Z}\) or \(\mathbb{R}\). The rules mirror those for equations: equal numbers may be added to or subtracted from both sides, and both sides may be multiplied or divided by the same positive number — but multiplying or dividing by a negative number reverses the inequality. Solutions over the real numbers are written as intervals such as \((-2,\infty)\) and shown on a number line, with an open circle for a strict inequality and a closed circle for \(\le\) or \(\ge\). Word problems — averages of marks, mixing acid solutions, consecutive odd integers, temperature conversion — are translated into inequalities and solved, illustrating wide use in science, statistics and economics.
Inequalities and their typesRules for solving inequalitiesLinear inequalities in one variableSolution over \(\mathbb{N}\), \(\mathbb{Z}\) and \(\mathbb{R}\)Representation on the number lineWord problems and applications
Key terms
Inequality
Two expressions related by \(<,\ >,\ \le\) or \(\ge\).
Strict vs slack
Strict inequalities use \(<\) or \(>\); slack (non-strict) use \(\le\) or \(\ge\).
Solution set
The set of all values of the variable that make the inequality true.
Sign reversal rule
Multiplying or dividing both sides by a negative number reverses the inequality.
Number-line representation
Open circle for a strict bound, closed circle for \(\le/\ge\).
Double inequality
A statement such as \(2\le x<6\) bounding the variable on both sides.
Extra questions & answers
Try one from this chapter
Multiplying both sides of an inequality by a negative number: