- 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
- 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
- Solve the following:
- 7m – 6 < 8m
- Subtract 7m from both sides: -6 < 8m – 7m
- -6 < m OR m > -6
- 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
- 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
- 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.