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CBSE Class 11 · Mathematics

Linear Inequalities

Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.

This lesson follows the official chapter “Linear Inequalities” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 11
  • Medium level
  • 8 concepts

1Inequalities

An inequality is a compare that stays open: <, >, ≤, ≥. 2x + 3 > 7 is not an equation. Solutions are a set (often a ray on the line), not one x unless the compare collapses. Flipping the sign when you multiply or divide by a negative is the school move.

Solving 2x+3>7 as x=2 is the equation-steal.

Figure. An inequality is a cut plus a side. x>2 takes every mark to the right of 2 and leaves 2 itself out — the open cut.

How it works

  1. See < or > (or the equal-or cousins)The compare.
  2. Treat it as a set of x, not one rootThe inequality.
  3. Remember the flip on a negative multiply/divideThe caution.

2Algebraic Solutions of Linear Inequalities in One Variable

Algebraic solution in one variable: collect x, divide, flip if needed. 2x + 3 > 7 → 2x > 4 → x > 2. The answer is the ray x > 2, drawn open at 2 if > (not ≥). A closed dot is for ≤ or ≥.

Leaving the answer as 2x>4 is not finished if x was asked.

Figure. Adding the same number keeps the sign. Dividing by a negative reverses it: -2x<6 becomes x>-3, then the open ray starts just after -3.

How it works

  1. Move the constantCollect.
  2. Divide; flip if the coefficient is negativeThe isolate.
  3. Draw the ray; open or closed as the symbolThe solution set.

2x + 3 > 7

Solve and describe the solution set.

  • Subtract 32x > 4
  • Divide by 2x > 2
  • Pictureopen at 2, arrow right

Pro tip. A ray, not a single root. No flip — 2 is positive.

3A definition is a test you can run

Inequality is a test: a compare plus a solution-set. If you only say “bigger”, you have a heading.

A number-line doodle with no open/closed is not the test.

Figure. A number is a solution only if substituting it makes a true sentence. For x<4, 3 works and 5 fails — the definition is that test, not a sketch.

How it works

  1. Name the compare and the rayThe object.
  2. Show the isolate and the pictureThe test.
  3. Then the word has contentThe definition ran.

4Name the given before the unknown

The given is the inequality. The unknown is the ray. Copy the sign of the coefficient before you divide.

Dividing −2x > 4 by −2 and keeping > is a silent flip-skip.

Figure. Before chasing x, name the coefficient, the constant term, the inequality sign and the right-hand number. The sign is the given that later decides whether the cut is open.

How it works

  1. Copy the writingThe given.
  2. Name the solution setThe unknown.
  3. Then isolate, flipping if neededGiven first.

5One worked case is enough at this class

One one-variable inequality is enough. A page of the same x>2 clone does not add a new flip.

Ten twins of an open dot are still one idea.

Figure. One school case is enough: 4x+1\le 13 subtracts 1, divides by 4, and graphs x\le 3 as a closed cut with the ray to the left.

How it works

  1. Solve one compareThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

6A miss: swapping the school name for the picture

A see-saw photo is a setting. The maths is a compare and a ray. A tilt is not x>2.

A balance-scale still is not ≥ unless you wrote it.

Figure. The miss is to treat the drawn cut as the answer. For x>3 the point 3 is only the fence; the solution is the open ray beyond it.

How it works

  1. Keep the see-saw as a storyA setting.
  2. Write the inequality and isolateThe maths.
  3. Then the rayDo not swap.

7Check by the opposite action or the opposite test

Check by substituting a point inside the ray and a point outside. For x>2, x=3 must work and x=1 must fail. The boundary 2 fails for > and works for ≥.

Checking only x=100 always “works” for many wrong rays.

Figure. Check by the opposite of solving: pick a number from the shaded side and one from the clear side. For x\le 2, 0 works and 4 fails.

How it works

  1. State x>2The claim.
  2. Try 3 and 1 (and 2 if the symbol needs it)The check.
  3. Mend if a inside-point failsHonest.

8Keep the claim at this chapter, not the next

This chapter’s claim stops at one-variable linear compares. It does not steal two-variable shading, and it does not start a linear-programming dump.

A page that already “shades a half-plane” here misses a later official idea if your book split it.

Figure. This chapter graphs one unknown on a line. A shaded half-plane is a two-variable picture and is not a claim this Class 11 chapter makes.

How it works

  1. Keep one x and a rayThis chapter.
  2. Leave two-variable pictures for when they are officialThe next map.
  3. Do not treat the map as already answeredClaim-size.
2x + 3 > 7 solves as
  1. x > 2 — a ray, open at 2
  2. x = 2
  3. x < 2

Isolate; a set, not a root.

Notes

  • The official chapter title is “Linear Inequalities”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • multiply/divide by a negative → flip the sign
  • x > a is a ray, open at a

Recap

Hold these pegs from the official chapter “Linear Inequalities”. The wording is ExamMaster’s teaching, not a textbook recap.

Inequalities
An inequality is a compare that stays open: <, >, ≤, ≥.
Algebraic Solutions of Linear Inequalities in One Variable
Algebraic solution in one variable: collect x, divide, flip if needed.
A definition is a test you can run
Inequality is a test: a compare plus a solution-set.
Name the given before the unknown
The given is the inequality. The unknown is the ray. Copy the sign of the coefficient before you divide.
One worked case is enough at this class
One one-variable inequality is enough. A page of the same x>2 clone does not add a new flip.
A miss: swapping the school name for the picture
A see-saw photo is a setting. The maths is a compare and a ray. A tilt is not x>2.

Practise Linear Inequalities

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