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CBSE · Class 11 · Mathematics

Linear Inequalities (Ch 5)

4 concepts · 32 practice questions with answers · Free to read

Inequalities and their rules

From daily life

A lift allows "at most 8 persons"; a bus ticket for children is "below 12 years". Real life uses "less than" and "more than" far more than "equal to".

Simple definition

An inequality compares two quantities with <, >, ≤ or ≥. Linear inequality in one variable has the form ax + b < 0 (or >, ≤, ≥). Rules: (1) adding or subtracting the same number on both sides keeps the sign; (2) multiplying or dividing by a positive number keeps the sign; (3) multiplying or dividing by a negative number reverses the sign.

Formula: a < b ⟹ a + c < b + c; a < b and c < 0 ⟹ ac > bc

Rule 1: Add and subtract — The direction stays the same.

  • Easy: x − 4 < 6 gives x < 10.
  • Medium: x + 7 ≥ 2 gives x ≥ −5.
  • Exam-type: 3 < 5 gives 3 + 2 < 5 + 2, i.e. 5 < 7.

Rule 2: Multiply or divide — Flip the sign when the multiplier is negative.

  • Easy: 2x < 10 gives x < 5.
  • Medium: −2x < 10 gives x > −5 (sign flips).
  • Exam-type: −x ≥ 3 gives x ≤ −3.
Remember

Test with numbers: 2 < 3 but −2 > −3, so multiplying by −1 flips the sign.

Common mistake

Forgetting to flip the sign after dividing by a negative number.

Memory clue

Negative number: Not the same sign — flip it.

Practice questions

Easy
  1. 1Solve x + 5 < 9.
  2. 2Solve 3x ≥ 12.
  3. 3Solve −x < 7.
Medium
  1. 4Solve −4x ≤ 20.
  2. 5Solve 2x − 3 > 7.
  3. 6If a > b, is −a > −b?
Challenging
  1. 7If x < −2, then 5 − 3x is greater than what number?
  2. 8Is it true that if a < b then 1/a > 1/b for positive a, b?
Show answers (try them yourself first)
  1. 1x < 4
  2. 2x ≥ 4
  3. 3x > −7
  4. 4x ≥ −5
  5. 5x > 5
  6. 6No — it flips: −a < −b.
  7. 7−3x > 6, so 5 − 3x > 11
  8. 8Yes — e.g. 2 < 4 and 1/2 > 1/4.

Solving a linear inequality

From daily life

You have ₹500 and each notebook costs ₹30. How many notebooks can you buy? The answer is not one number but a whole range.

Simple definition

Solve like an equation: collect x on one side, numbers on the other, divide by the coefficient (flip the sign if it is negative). The answer is a range of values, drawn on the number line with an open circle for < or > and a filled circle for ≤ or ≥.

-3-2-10123456►

In the figure: x > 2: open circle at 2 and the line going right.

Rule 1: One-step and two-step — Do the same on both sides.

  • Easy: 3x − 5 ≤ 7 gives 3x ≤ 12, x ≤ 4.
  • Medium: 5 − 2x < 1 gives −2x < −4, x > 2.
  • Exam-type: 4(x − 1) ≥ 2x + 6 gives 2x ≥ 10, x ≥ 5.

Rule 2: With fractions — Multiply by the common denominator (positive).

  • Easy: x/3 + 2 < 4 gives x < 6.
  • Medium: (x − 1)/2 ≥ (x + 2)/3 gives 3x − 3 ≥ 2x + 4, x ≥ 7.
  • Exam-type: 2x/5 − 1 > 3 gives x > 10.
Remember

Check your answer by choosing one number from the solution and one outside.

Common mistake

Using a filled circle for < or >.

Memory clue

Open circle = not allowed, filled = allowed.

Practice questions

Easy
  1. 1Solve 2x + 3 < 11.
  2. 2Solve 7 − x ≥ 2.
  3. 3Solve 3(x + 2) > 12.
Medium
  1. 4Solve (2x − 1)/3 ≤ 5.
  2. 5Solve 5x − 2 < 3x + 8.
  3. 6Solve x/2 − x/3 ≥ 1.
Challenging
  1. 7Solve 6 − 4x > 2(x − 3).
  2. 8Solve (x + 1)/2 < (2x − 3)/3.
Show answers (try them yourself first)
  1. 1x < 4
  2. 2x ≤ 5
  3. 3x > 2
  4. 42x − 1 ≤ 15, x ≤ 8
  5. 52x < 10, x < 5
  6. 6x/6 ≥ 1, x ≥ 6
  7. 76 − 4x > 2x − 6, 12 > 6x, x < 2
  8. 83x + 3 < 4x − 6, so x > 9

Replacement sets and graph of the solution

From daily life

"Number of notebooks" can only be whole numbers, but "length of a rope" can be any real. The same inequality gives different answer sets.

Simple definition

The solution depends on the replacement set (the allowed values of x): natural numbers N, integers Z or real numbers R. First solve in R, then list or draw only the allowed values.

-3-2-10123456

In the figure: −2 ≤ x ≤ 3: filled circles at both ends of the segment.

Rule 1: Same inequality, different sets — Solve first, then pick.

  • Easy: x < 4: in N the solution is {1, 2, 3}.
  • Medium: x < 4: in Z it is {…, 0, 1, 2, 3}.
  • Exam-type: x < 4: in R it is the interval (−∞, 4).

Rule 2: Double inequality — Work on all three parts at once.

  • Easy: −3 < 2x + 1 ≤ 9 gives −4 < 2x ≤ 8, so −2 < x ≤ 4.
  • Medium: 1 ≤ 3x − 2 < 7 gives 3 ≤ 3x < 9, so 1 ≤ x < 3.
  • Exam-type: −6 ≤ 4 − 2x < 2 gives −10 ≤ −2x < −2, so 1 < x ≤ 5.
Remember

Write the solution as an interval or a number-line graph; both are accepted.

Common mistake

Listing real numbers one by one when the replacement set is R.

Memory clue

Solve in R, then filter by the set.

Spot the difference

x ∈ Nx ∈ R
Counting numbers onlyEvery point on the line
Dots on the lineA continuous segment

Practice questions

Easy
  1. 1Solve x + 2 < 6 for x ∈ N.
  2. 2Solve 2x − 1 ≤ 5 for x ∈ Z, x ≥ 0.
  3. 3Solve −2 < x ≤ 3 for x ∈ Z.
Medium
  1. 4Solve −5 ≤ 2x − 1 < 7 and write as an interval.
  2. 5How many natural numbers satisfy 3x − 4 ≤ 20?
  3. 6Solve 2 < (3x − 4)/2 ≤ 7.
Challenging
  1. 7Find the integers x with 5 < 2x + 1 < 15.
  2. 8Find all prime numbers p satisfying 2p − 3 < 15.
Show answers (try them yourself first)
  1. 1x < 4: {1, 2, 3}
  2. 2x ≤ 3: {0, 1, 2, 3}
  3. 3{−1, 0, 1, 2, 3}
  4. 4−4 ≤ 2x < 8, so −2 ≤ x < 4, i.e. [−2, 4)
  5. 5x ≤ 8: 8 natural numbers
  6. 64 < 3x − 4 ≤ 14, 8 < 3x ≤ 18, 8/3 < x ≤ 6
  7. 72 < x < 7: {3, 4, 5, 6}
  8. 8p < 9: {2, 3, 5, 7}

Inequalities in word problems

From daily life

Budgets, speed limits, marks to pass: they all give "at least" or "at most" conditions.

Simple definition

Translate words into symbols: "at least" means ≥; "at most" means ≤; "more than" means >; "less than" means <. Let the unknown be x, write the condition, then solve.

Rule 1: Budget — Total cost is at most the money you have.

  • Easy: A pen costs ₹12. With ₹100, 12x ≤ 100 gives x ≤ 8, so at most 8 pens.
  • Medium: A taxi costs ₹40 plus ₹15 per km; with ₹250: 40 + 15x ≤ 250 gives x ≤ 14 km.
  • Exam-type: To get an average of at least 60 in 3 tests with 55 and 62, the third score y satisfies (117 + y)/3 ≥ 60, so y ≥ 63.

Rule 2: Number problems — Form the expression carefully.

  • Easy: Three times a number decreased by 4 is more than 11: 3x − 4 > 11, x > 5.
  • Medium: The sum of two consecutive odd numbers is more than 40 and the smaller is x: 2x + 2 > 40, x > 19.
  • Exam-type: A rectangle's length is 3 m more than its width x and its perimeter is at most 50 m: 4x + 6 ≤ 50, x ≤ 11.
Remember

Always check that the answer makes sense (a count cannot be negative or fractional).

Common mistake

Choosing ≤ when the problem says "less than".

Memory clue

At least = floor (≥). At most = ceiling (≤).

Practice questions

Easy
  1. 1A box holds at most 20 kg. Each book weighs 1.5 kg. Write an inequality for the number of books x.
  2. 2A student needs at least 40 marks. She has 28. Write an inequality for the marks y she still needs.
  3. 3Solve 2x + 5 < 19 and say how many natural numbers fit.
Medium
  1. 4A phone plan costs ₹200 plus ₹2 per message. For how many messages is the bill at most ₹500?
  2. 5The marks in two tests are 62 and 71. How many marks are needed in the third test for an average of at least 70?
  3. 6Find the largest integer x with 4x − 7 < 25.
Challenging
  1. 7A rectangle has width 5 m and a perimeter of less than 40 m. What lengths are possible?
  2. 8Find two consecutive even numbers whose sum is more than 30 with the smaller x; list the smallest pair.
Show answers (try them yourself first)
  1. 11.5x ≤ 20
  2. 2y ≥ 12
  3. 3x < 7: 6 natural numbers
  4. 4200 + 2x ≤ 500, x ≤ 150
  5. 5133 + y ≥ 210, y ≥ 77
  6. 6x < 8, so 7
  7. 72(l + 5) < 40, l < 15, so 0 < l < 15
  8. 82x + 2 > 30, x > 14, smallest even x is 16: (16, 18)