Linear Inequalities — Class 11 CBSE Mathematics revision notes
A 2-page visual revision sheet for Linear Inequalities, written for Class 11 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Linear Inequalities in one look
solving with < and > instead of =, one variable at a time and then two
What an inequality is
- a statement comparing two expressions with <, >, ≤ or ≥ instead of =
- a strict inequality (<, >) excludes the boundary value. a slack one (≤, ≥) includes it
- solving one means finding every value of the variable that makes the statement true, not just one value
- the solution to a linear inequality in one variable is usually a whole ray or interval, never a single number
Rules for solving, same as an equation with one exception
- add or subtract the same number from both sides
a<b ⇒ a+c<b+c - multiply or divide both sides by a positive number
a<b, c>0 ⇒ ac<bc - multiply or divide both sides by a negative number
a<b, c<0 ⇒ ac>bc
Does the sign flip
what are you multiplying or dividing both sides by
- a positive number — sign stays the same: 3x<12 divided by 3 stays x<4
- a negative number — sign flips: -3x<12 divided by -3 becomes x>-4, not x<-4
A compound inequality
Q. solve -5 ≤ 2x-3 < 7 for real x
- add 3 to every part: -5+3 ≤ 2x-3+3 < 7+3, giving -2 ≤ 2x < 10
- divide every part by 2, a positive number so the signs stay put: -1 ≤ x < 5
Answer: x is any real number with -1 ≤ x < 5
Inequalities in two variables
- a linear inequality like 2x+3y≤ 6 is satisfied by a whole region of the xy-plane, not a curve
- the boundary line 2x+3y=6 splits the plane into two half-planes. every point in one half satisfies the inequality, every point in the other doesn't
- a strict inequality (< or >) draws the boundary as a dashed line, since points on it are excluded. ≤ or ≥ draws it solid
- test any one point not on the line, usually (0,0), to see which half-plane is the solution
Page 1 of 2. The full sheet, all 2 pages, opens free in the app.
What this Linear Inequalities sheet covers
- What an inequality is
- Rules for solving, same as an equation with one exception
- add or subtract the same number from both sides
- multiply or divide both sides by a positive number
- multiply or divide both sides by a negative number
- Does the sign flip
- A compound inequality
- Inequalities in two variables
- A system of two inequalities
- Reading a word problem into an inequality
- A word problem, translated and solved
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Linear Inequalities the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.