Lesson 1 of 10 · 7 min
Inequalities and their kinds
NCERT §5.1, §5.2
Imagine the class running a juice stall at the school fête. Riya has ₹500 to spend on fruit cartons that cost ₹45 each. She does not need to spend exactly ₹500; she only must not spend more. That is a job for an inequality, not an equation.
The story this chapter follows: Juice stall at the school fête
The lesson in notes
In short
An inequality is a statement that compares two numbers or two expressions with one of the signs < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to).
Many real situations give inequalities rather than equations. With ₹500 to spend on fruit cartons at ₹45 each, the number x of cartons must satisfy 45x ≤ 500, because the bill may be less than the money in hand but not more.
The sign ≤ combines two statements: 45x ≤ 500 means 45x < 500 or 45x = 500. Likewise ≥ means 'greater than or equal to'.
Numerical inequalities compare fixed numbers, such as 4 < 9 or −2 > −5. Literal inequalities contain a variable, such as x < 7 or y ≥ −1.
A double inequality puts one quantity between two others, as in 4 < t < 10, read as 't is greater than 4 and less than 10'.
Strict inequalities use < or > and exclude the boundary; slack inequalities use ≤ or ≥ and include it.
ax + b < 0, ax + b ≥ 0 and similar forms with a ≠ 0 are linear inequalities in one variable. ax + by ≤ c with a ≠ 0 and b ≠ 0 is linear in two variables: for mango cartons at ₹45 and orange cartons at ₹30 within ₹500, 45x + 30y ≤ 500.
An inequality such as x² − 5x + 6 > 0 is not linear; it is quadratic. This chapter works with linear inequalities in one variable.