Popular Courses
Beginner – Linear Regression
Introduction
Beginner – Decision Trees
Introduction
Beginner – Neural Networks
Introduction

Sign in to Curvingo

or

New Account     Reset Password     

Plans

Details
Linear Inequalities
Introduction
Inequalities
Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
Practice: Solving Linear Inequalities
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
Curvingo · Made in India

Share

Send us Feedback

Please let us know if you have any feedback and suggestions, or if you find any errors and bugs in our content.

Sorry, your message couldn’t be submitted. Please try again!

Thanks for your feedback!

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

Linear InequalitiesInequalities

Reading time: ~20 min

Two shopping trips show where inequalities come from.

(i) Ravi goes to the market with ₹200 to buy rice. The shop sells it only in 1 kg packets, at ₹30 a packet. How many packets can he take home?

Fill Ravi's bag. Add packets one at a time until the shopkeeper says no.

Call the number of packets x. Ravi then spends 30x rupees, and that must stay below the ₹200 in his purse:

30x < 200 \qquad \ldots (1)

Statement (1) is not an equation: it has no sign of in it. And notice why it says "less than" and not "equals". Packets come only whole, so the bill jumps 0, 30, 60, …, 180, 210 — it skips 200 altogether, because 200 is not a of 30. The best Ravi can do is 6 packets for ₹180, with ₹20 he simply cannot use.

Reshma's Stationery Run

(ii) Reshma has ₹120 to spend on registers and pens. A register costs ₹40 and a pen costs ₹20. If she buys x registers and y pens, she spends ₹(40x + 20y) — and that can be anything up to the full ₹120.

Price some carts in the table: each square is one cart. Complete all three missions.

Every blue or green cart obeys one rule:

40x + 20y \le 120 \qquad \ldots (2)

The sign \le packs two statements into one. The blue carts, which leave change, satisfy

40x + 20y < 120 \qquad \ldots (3)

and the green carts, which use every rupee, satisfy

40x + 20y = 120 \qquad \ldots (4)

Statement (4) is an equation, while statement (3) is an .

Three Kinds of Inequality

Here is the precise idea behind statements (1), (2) and (3).

Definition 1 Two real numbers or two algebraic expressions related by the <, >, \le or \ge form an inequality.

Inequalities come in a few flavours. Name each one.

3 < 5 and 7 > 5 compare two fixed numbers. They are inequalities.

x < 5, y > 2, x \ge 3 and y \le 4 contain a letter standing for a number. They are inequalities.

3 < 5 < 7, 3 \le x < 5 and 2 < y < 4 use two signs at once. They are inequalities.

In the language of the glossary: the first kind are numerical inequalities, the second literal inequalities and the third double inequalities. Read a double inequality from the middle outwards: 3 \le x < 5 says "x is greater than or equal to 3 and less than 5", and 3 < 5 < 7 says "5 is greater than 3 and less than 7".

Strict, Slack, Linear

Here are some more inequalities, sorted by how many variables they contain. In each, a, b and c stand for fixed real numbers.

One variable, a \neq 0Two variables, a, b \neq 0
ax + b < 0 \quad (5)ax + by < c \quad (9)
ax + b > 0 \quad (6)ax + by > c \quad (10)
ax + b \le 0 \quad (7)ax + by \le c \quad (11)
ax + b \ge 0 \quad (8)ax + by \ge c \quad (12)

and two that contain x^2:

ax^2 + bx + c \le 0 \quad (13) \qquad ax^2 + bx + c > 0 \quad (14)

Inequalities (5), (6), (9), (10) and (14) use only < or >, so the boundary value is never allowed. They are — strict inequalities. Inequalities (7), (8), (11), (12) and (13) allow equality too; they are slack inequalities.

Inequalities (5) to (8) are linear inequalities in variable, x, and (9) to (12) are linear inequalities in variables, x and y. Inequalities (13) and (14) are not at all: because of the x^2 term they are quadratic inequalities in one variable.

In this chapter we stay with linear inequalities in variable.

Speed Round: Linear or Not?

Simplify in your head if you need to, then sort the inequality. After each answer you will see why.

Get five right to move on.

Sina