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Inequalities

• 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.
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