Definition 1
Two real numbers or two algebraic expressions related by the symbol <, >, ≤, or ≥ form an inequality. Click a card to see it plotted on a number line.
Numerical strict
3 < 5, 7 > 5
Both sides are plain numbers — no variable at all.
Literal strict
x < 5, y > 2
A variable appears on one side — the value of x/y decides truth.
Literal slack
x ≥ 3, y ≤ 4
"Slack" inequalities allow equality — note the underline in ≤ / ≥.
Double inequality
3 < x < 5
Read: "x is greater than 3 and less than 5" — two conditions at once.
Double inequality
3 ≤ x < 5
Mixed brackets: closed at 3, open at 5.
Double inequality
2 < y ≤ 4
Open at 2, closed at 4.
Selected example on the number line
General forms — which are linear?
The chapter names ten general forms. Toggle to see which are strict vs. slack, and which are truly linear (degree 1) versus quadratic.
Rule 1 & Rule 2 — keeping the balance
Start from 3x < 7. Apply an operation to both sides and watch whether the inequality sign survives — or flips.
Operate on both sides
Rule 1 — add/subtract the same number on both sides: the sign never changes.
Rule 2 — multiply/divide both sides by the same number: sign stays if the number is positive, and reverses if it is negative.
The balance scale
The heavier pan is the greater side. Watch what happens to the tilt as you apply operations on the left.
Solve a linear inequality in one variable
Build an inequality of the form a₁x + b₁ [op] a₂x + b₂, choose the number system, and watch it get solved step by step (Rule 1 then Rule 2) with the solution drawn on a number line — just like Examples 1–6.
Build your inequality
Number line
A filled ray shows a continuous real solution set. Dots show discrete natural-number / integer solutions.
Double (compound) inequalities
Solve two conditions simultaneously, like −8 ≤ 5x − 3 < 7 (Example 9). Enter lo [op1] ax + b [op2] hi and every step is applied to all three parts at once.
Build it
Number line
System of two inequalities
Like Example 11: solve a₁x + b₁ [op1] c₁ and a₂x + b₂ [op2] c₂ separately, then find where both hold — the overlap on the number line.
Inequality 1
Inequality 2
Each box solves a·x + b [op] c. Example 11's 3x − 7 < 5 + x simplifies first to 2x < 12 (defaults above); its second condition 11 − 5x ≤ 1 is entered directly. Try your own numbers too.
Number lines
Inequality 1
Inequality 2
Combined (both true)
Linear inequalities in two variables
A statement like Reshma's 40x + 20y ≤ 120 is a region, not a line. Set a, b, c and the sign for ax + by [op] c and see the half-plane it describes.
Build the inequality
Boundary is dashed for strict (<, >) and solid for slack (≤, ≥) — it's included in the solution only when solid.
Registers × pens: Reshma has ₹120, registers cost ₹40, pens cost ₹20 — every point in the shaded region (with x,y ≥ 0) is an affordable combination.
Plane
Word problems
Three classic inequality set-ups from the chapter, made interactive.
Average marks Ex. 7 & 21
Marks in earlier tests are fixed; find the minimum mark needed in the remaining test(s) for a target average.
Acid mixture Ex. 13
Mix a strong acid solution into a weaker one so the final concentration lands strictly between two bounds.
Consecutive integers Ex. 8
Find all pairs of consecutive odd/even natural numbers, both larger than a lower bound, whose sum stays under an upper bound.
Temperature conversion range Ex. 12
A Celsius range converts to a Fahrenheit range using C = 5/9 (F − 32).