• Common inequality symbols and their meanings are:
  • ≠ is not equal to
  • > is greater than
  • < is less than
  • ≥ is greater than or equal to
  • ≤ is less than or equal to
• If you alternate the RHS with the LHS or the LHS with the RHS, the direction of inequality would change. For example:
  • In 9x > 7
  • When you interchange the RHS with the LHS, the result would be: 7 < 9x
  • NOTE: 9x > 7 is the same as 7 < 9x but not equal to 7 > 9x and 9x < 7
• When multiplying or dividing through by a negative number, the direction of the inequality changes. For example:
  • In -9x > 27
  • When you multiply through by -2, the result would be: 18x < -54
    • NOTE: If you multiplied by 2, the result would be -18x > 54
  • When you divide through by -9, the result would be:  x < -3
    • NOTE: If you divide by 9, the result would be: -x > 3

Examples

1. Solve 19 ≥ 4 – 5x
   • Subtract 4 from both sides: 15 ≥ -5x
   • Divide both sides by -5: ¹⁵/_-5 ≤ ^-5x/_-5
   • ∴ -3 ≤ x or x ≥ -3
2. Solve the following:
   1. 7m – 6 < 8m
      • Subtract 7m from both sides: -6 < 8m – 7m
      • -6 < m OR m > -6
   2. 2x + 5(x – 3) > x + 9
      • Open the bracket: 2x + 5x – 15 > x + 9
      • 7x – 15 > x + 9
      • Subtract x from both sides: 6x – 15 > 9
      • Add 15 to both sides: 6x > 24
      • DIvide through by 6: x > 4
   3. ¹/_{(x + 1)} > ^-1/₃
      • Multiply both sides by (x + 1): 1 > ^-1/₃(x + 1)
      • Multiply through by 3: 3 > -1(x + 1)
      • Open the bracket: 3 > -x – 1
      • Add 1 to both sides: 4 > -x
      • Divide through by -1: -4 < x OR x > -4
3. A triangle has sides of x cm, (x + 4) cm, and 11 cm, where x is a whole number in cm. If the perimeter of the triangle is less than 32 cm, find the possible values of x.
   • Perimeter of the triangle = x + (x + 4) + 11
   • ⇒ x + (x + 4) + 11 < 32
   • 2x + 15 < 32
   • 2x < 17
   • ∴ x < 8¹/₂
   • Also, in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
   • ⇒ x + (x + 4) > 11
   • 2x + 4 > 11
   • 2x > 7
   • ∴ x > 3¹/₂
   • ⇒ x < 8¹/₂ and x > 3¹/₂
   • But x must be a whole number in cm; thus, the possible values of x are 4, 5, 6, 7, or 8.
   • Check:
     • when x = 4, the perimeter = 4 + 8 + 11 = 23 cm
     • when x = 8, the perimeter = 8 + 12 + 11 = 31 cm
     • The lowest and highest values of x have been checked. The perimeters in both cases are less than 32 cm, there is no need to check the other values.