“Hard times don’t create heroes. It is during the hard times when the ‘hero’ within us is revealed.” – Bob Riley

# Lecture - 1 Chapter 6 Linear Inequalities

Basics of Inequalities, Strict and Slack inequalities

Working with inequalities: Adding positive or negative numbers Subtracting positive or negative numbers Multiplying by positive or negative numbers Dividing by positive or negative numbers

Representing Real numbers in form of open intervals and closed intervals

NCERT Exercise 6.1 |

**Question 1.** Solve \( 24x < 100 \), when

**(i)** *x* is a natural number.**(ii)** *x* is an integer.

**Question 2.** Solve – 12*x* > 30, when **(i)** *x* is a natural number.**(ii)** *x* is an integer.

**Question 3.** Solve 5*x* – 3 < 7, when **(i)** *x* is an integer.**(ii)** *x* is a real number.

**Question 4.** Solve 3*x* + 8 >2, when **(i)** *x* is an integer.**(ii)** *x* is a real number.

Solve the inequalities in Exercises 5 to 16 for real *x*.

**Question 5.** \(4x+3<5x+7\)

**Question 6.** \(3x-7>5x-1\)

**Question 7.** \(3(x-1) \le 2 (x-3)\)

**Question 8.** \(3(2-x) \ge 2(1-x)\)

**Question 9.** \(x + \frac{x}{2}+\frac{x}{3}<11\)

**Question 10.** \(\frac{x}{3}>\frac{x}{2}+1\)

**Question 11.** \(\frac{3(x-2)}{5} \le \frac{5(2-x)}{3}\)

**Question 12.** \(\frac{1}{2} \left ( \frac{3x}{5} + 4 \right ) \ge \frac{1}{3} (x-6)\)

**Question 13.** \(2 (2x + 3) – 10 < 6 (x – 2)\)

**Question 14.** \(37 – (3x + 5) > 9x – 8 (x – 3)\)

**Question 15.** \(\frac{x}{4}< \frac{(5x-2)}{3} – \frac{(7x-3)}{5}\)

**Question 16.** \(\frac{(2x-1)}{3} \ge \frac{(3x-2)}{4} – \frac{(2-x)}{5}\)

Solve the inequalities in Exercises 17 to 20 and show the graph of the solution in each case on number line

**Question 17.** \(3x-2<2x+1\)

**Question 18.** \(5x-3 \ge 3x-5\)

**Question 19.** \(3(1-x) < 2(x+4)\)

**Question 20.** \(frac{x}{2} \ge \frac{(5x-2)}{3} – \frac{(7x-3)}{5}\)