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 \((11,13),(13,15),(15,17),(17,19)\).

CBSE · Senior secondary

CBSE Class 11 — Notes, Chapters & Practice Quizzes

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 \((11,13),(13,15),(15,17),(17,19)\).

Master every Class 11 chapter — the base your board marks and your NEET / JEE / CUET prep both stand on.

13 subjects 171 chapters 177 practice quizzes NCERT syllabus

66 quiz attempts so far Dual AI-verified questions 10 quizzes free, then 10 credits

Notice Board

Students preparing for CBSE Class 11 Annual Assessment

View all

Formula of the day

Chemistry · Gaseous State · 28 formulae in the cheatsheet

Trap of the day

In v² = u² + 2as, a is negative for deceleration — keep the sign or the stopping distance comes out wrong.

Chapter 5: Linear Inequalities — Class 11 Mathematics

Mathematics · 14 chapters

Summary, key terms, important questions and a practice quiz with AI diagnosis for each.

Chapter 5: Linear Inequalities

Try one from this chapter

Multiplying both sides of an inequality by a negative number:

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.

Important questions

Explore interactively

Key-term flashcards

Flip cards · mark known · keyboard ← → and Space

6 cards

Term1 / 6

Inequality

Tap to reveal

Meaning1 / 6

Two expressions related by \(<,\ >,\ \le\) or \(\ge\).

Tap to flip back

Tap card to flip

Keyboard: ← → to move · Space to flip

Practice quiz · Linear Inequalities

Score on this chapter, climb the leaderboard, and get an AI diagnosis of your mistakes.

Dual AI-verified questions Real exam pattern 2 quizzes free, then 10 credits per quiz

Linear Inequalities

Maths 10 Qs · ~10 min