On this page
Linear inequalities look easy until a negative sign appears. Most lost marks in this chapter come from one rule forgotten at the wrong moment, or from an open circle drawn where a filled one belongs. This post sets out what the Class 11 syllabus covers, works through each type of question with checked answers, and ends with a practice routine.
What does the Class 11 syllabus include?
The CBSE Mathematics curriculum for Class XI (2026-27) lists "Algebraic solutions of linear inequalities in one variable and their representation on the number line" under Algebra, a unit worth 25 of 80 marks. A separate table in the same document says that the graphical solution of linear inequalities in two variables, and of systems of them, is included but "will be assessed only formatively", so it does not add to the year-end exam. The current NCERT chapter on linear inequalities covers only the one-variable work. Check the curriculum for your own year, as these notes change.
What are the rules for solving an inequality?
NCERT gives two rules:
You can add or subtract the same number on both sides without changing the sign.
You can multiply or divide both sides by the same positive number. If you multiply or divide by a negative number, the sign reverses: < becomes >, ≤ becomes ≥.
Why the flip? Because 3 > 2, but −3 < −2. Multiplying by −1 swaps the order on the number line.
Example 1. Solve 3(x − 2) ≥ 5x + 4 for real x.
Expand: 3x − 6 ≥ 5x + 4.
Subtract 5x and add 6: −2x ≥ 10.
Divide by −2 and flip: x ≤ −5.
The solution set is (−∞, −5]. Check with x = −6: 3(−8) = −24 and 5(−6) + 4 = −26, and −24 ≥ −26 is true.
Example 2. Solve (2x − 1)/3 − (x + 2)/4 > 1.
Multiply by 12, which is positive, so no flip: 4(2x − 1) − 3(x + 2) > 12.
Expand: 8x − 4 − 3x − 6 > 12, so 5x − 10 > 12.
5x > 22, so x > 22/5, or x > 4.4.
How do you show the solution on a number line?
For < or >, draw an open (hollow) circle at the boundary: the point itself is not included.
For ≤ or ≥, draw a filled circle: the point is included.
Shade the side that holds the solutions.
For Example 1, put a filled circle at −5 and shade to the left. For Example 2, put an open circle at 4.4 and shade to the right.
How do you solve double inequalities and systems?
Example 3. Solve −7 < 2 − 3x ≤ 8.
Subtract 2 from all three parts: −9 < −3x ≤ 6.
Divide all three parts by −3 and flip both signs: 3 > x ≥ −2.
Write it in increasing order: −2 ≤ x < 3, or [−2, 3).
Example 4. Solve the system 2x + 1 > x − 3 and 3x − 5 ≤ 7.
The first gives x > −4. The second gives 3x ≤ 12, so x ≤ 4.
The solution is the overlap: −4 < x ≤ 4. On the number line, open circle at −4, filled circle at 4, shade between.
Example 5 (integers). Solve 4x − 3 < 18 when x is a natural number. Then 4x < 21, so x < 5.25, and the solution set is {1, 2, 3, 4, 5}. Always read whether the question wants natural numbers, integers or real numbers.
Example 6 (word problem). Arjun scored 58 and 71 in two unit tests out of 100. What must he score in the third to average at least 65? Let the score be x. Then (58 + 71 + x)/3 ≥ 65, so 129 + x ≥ 195 and x ≥ 66. Since the test is out of 100, the answer is 66 ≤ x ≤ 100.
What about inequalities in two variables?
You need them for Class 12 linear programming, even though they are formatively assessed in Class 11. Take 2x + y ≤ 6. Draw the line 2x + y = 6 through (3, 0) and (0, 6), solid because of the ≤. Test the origin: 0 ≤ 6 is true, so shade the side that contains (0, 0). For a strict < you would draw the line dashed. Our guide to linear programming for CBSE builds on exactly this.
What mistakes cost marks?
Not flipping when dividing by a negative, especially in double inequalities where both signs must flip.
Multiplying by an expression containing x, such as x − 2, whose sign you do not know.
Open and filled circles swapped.
Writing 3 > x ≥ −2 and stopping. Rewrite it in increasing order and as an interval.
Ignoring the number set. Natural numbers, integers and reals give different answers.
A practice routine for this chapter
Mixed practice, where you have to decide which method a question needs, has been shown to help. In a study of 126 seventh-grade students, Rohrer and colleagues found that interleaved practice beat practising one type at a time on tests a day and a month later.
Day 1: Do Exercise 5.1, questions 5 to 16, checking each answer with one test point.
Day 2: Do six double inequalities and systems from the miscellaneous exercise, drawing each on a number line.
Day 4: Make a mixed set of eight: two of each type above, shuffled.
Day 7: Redo every question you got wrong, without notes.
When you cannot see why your answer is wrong, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. For the set notation used in interval answers, see our guide to sets and relations in Class 11.
Start now: solve Example 3 yourself on paper, then test x = 0 in the original inequality.
