Linear Inequalities Class 11 Notes - Chapter 6

Two algebraic expressions or real numbers related by the symbol ‘>’, ‘<’, ‘≥’, or ‘≤’ forms an inequality. Statements such as 4x + 2y ≤ 20, 9x + 2y ≤ 720, 9x + 8y ≤ 70 are inequalities. 4 < 6; 8 > 6 are the examples of numerical inequalities. x < 6; y > 3; x ≥ 4, y ≤ 5 are examples of literal inequalities. 4 < 6 < 8, 4 < x < 6 and 3 < y < 5 are examples of double inequalities.

Rules for solving linear equations

  • Rule 1: The same number can be subtracted or added from both the sides (LHS and RHS) of an equation.
  • Rule 2: Both LHS and RHS of an equation can be divided or multiplied by the same non-zero number.

For solving the inequalities we follow the same rules except with a difference that the sign of inequality is reversed (< to > and ≤ to ≥) whenever an inequality is divided or multiplied by a -ve number. Some examples of inequalities are:

  • Strict inequalities: ax + b < 0, ax + b > 0, ax + by < c, \(ax^{2}+bx+c>0\)
  • Slack inequalities: ax + b ≤ 0, ax + b ≥ 0, ax + by ≤ c, ax + by ≥ c, \(ax^{2}+bx+c\leq 0\)
  • Linear inequalities in one variable: ax + b < 0 ax + b > 0 ax + b ≤ 0 ax + b ≥ 0 [when a ≠ 0]
  • Linear inequalities in two variable: ax + by < c, ax + by > c, ax + by ≤ c, ax + by ≥ c.

Rules for solving an Inequality

  • We can add or subtract the same number to LHS and RHS of inequality without changing the sign of that inequality.
  • We can divide or multiply both sides of an inequality by the same positive number.
  • The sign of the inequality is reversed when both sides (LHS and RHS) are divided or multiplied by the same negative number.

Find the solutions of the given equation 8x + 6 < 12x + 14.

Solution: We have, 8x + 6 < 12x + 14

or 8x -12x < 14 – 6

or -4x < 8

or -x < 2

or x > -2 All the real numbers greater than -2 are the solutions of the given inequality. Hence, the set of solution is (–2, ∞).

Linear Inequalities Practice Questions

  1. Solve the inequalities (i) 4 ≤ 6y – 8 ≤ 10 (ii) 12 ≤ -6 (4y – 8) < 24
  2. Represent the solutions of the system of inequalities on the number line. 6x – 14 < 10 + 2x, 22 – 10x ≤ 2.
  3. Solve -16 ≤ 10y – 6 < 14.
  4. Using the graphical method, solve the following system of inequalities:
  • x + 2y ≤ 8
  • 2x + y ≤ 8
  • x > 0, y > 0
  1. Solve 14x + 6 < 10x + 18. Also represent the solutions on the number line.

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