What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

What is a linear inequality in one variable?

A statement of the form \(ax+b ,\le,\ge\)), \(a\ne0\), relating a linear expression to zero by an inequality sign.

Solve \(5x-3<3x+1\) for real \(x\).

\(5x-3<3x+1\Rightarrow 2x<4\Rightarrow x<2\); the solution set is \((-\infty,2)\).

Solve \(4x+3<6x+7\) for real \(x\).

\(4x+3 -2\); the solution set is \((-2,\infty)\).

Why does dividing by a negative reverse the sign?

Because \(3>2\) but \(-3<-2\): multiplying both sides by a negative number flips the order, so the inequality direction must reverse.

How is \(x<3\) shown on a number line?

Place an open (hollow) circle at \(3\) and shade the line to the left, since \(3\) itself is excluded.

Solve \(7x+3<5x+9\) and give the interval.

\(2x<6\Rightarrow x<3\); the solution interval is \((-\infty,3)\).

A student scored 62 and 48 in two terms. What minimum third-term mark gives an average of at least 60?

\(\dfrac{62+48+x}{3}\ge 60\Rightarrow 110+x\ge180\Rightarrow x\ge70\); he needs at least 70 marks.

Solve the double inequality \(-8\le 5x-3<7\).

Add 3: \(-5\le 5x<10\); divide by 5: \(-1\le x<2\).

Find consecutive odd natural numbers \(>10\) whose sum is \(<40\).

With smaller number \(x>10\) and \(x+(x+2)<40\Rightarrow x<19\); so \(10<x<19\), giving pairs \((11,13),(13,15),(15,17),(17,19)\).

Linear Inequalities

Q: State the two key rules for solving inequalities.

Equal numbers may be added to or subtracted from both sides; both sides may be multiplied or divided by the same positive number, but by a negative number the sign reverses.

Q: Solve \(5x-3<3x+1\) for real \(x\).

\(5x-3<3x+1\Rightarrow 2x<4\Rightarrow x<2\); the solution set is \((-\infty,2)\).

Q: Solve \(4x+3<6x+7\) for real \(x\).

\(4x+3 -2\); the solution set is \((-2,\infty)\).

Q: Why does dividing by a negative reverse the sign?

Because \(3>2\) but \(-3<-2\): multiplying both sides by a negative number flips the order, so the inequality direction must reverse.

Q: How is \(x<3\) shown on a number line?

Place an open (hollow) circle at \(3\) and shade the line to the left, since \(3\) itself is excluded.

Q: Solve \(7x+3<5x+9\) and give the interval.

\(2x<6\Rightarrow x<3\); the solution interval is \((-\infty,3)\).

Q: A student scored 62 and 48 in two terms. What minimum third-term mark gives an average of at least 60?

\(\dfrac{62+48+x}{3}\ge 60\Rightarrow 110+x\ge180\Rightarrow x\ge70\); he needs at least 70 marks.

Q: Solve the double inequality \(-8\le 5x-3<7\).

Add 3: \(-5\le 5x<10\); divide by 5: \(-1\le x<2\).

Q: Find consecutive odd natural numbers \(>10\) whose sum is \(<40\).

With smaller number \(x>10\) and \(x+(x+2)<40\Rightarrow x<19\); so \(10<x<19\), giving pairs \((11,13),(13,15),(15,17),(17,19)\).

Class 11 Mathematics · Chapter 5

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Every formula of Ch 1–14, with its condition

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25 marks: Unit II, Ch 4–8
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CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

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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:

Flashcards & interactive practice