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How Do You Solve a Multi-Step Inequality Using Reverse Order of Operations?

Solve this inequality for g:
((30-4g)/3) ≥ 6

Summary

  1. We want to get 'g' on one side by itself
  2. Multiplying by 3 is the same as multiplying by 3/1

Notes

    1. Solving for 'g' just means we want to get it by itself on one side of the inequality
    1. In order to get 'g' alone, we need to perform the order of operations in reverse
    1. So ask yourself what the opposite operation of dividing by '3' is
    1. Multiplying by 3 is the opposite of dividing by '3'!
    2. This will help us get closer to solving for 'g'
    1. Multiplying by 3 is the opposite of dividing by '3'!
    2. This will help us get closer to solving for 'g'
    3. (3/1)•((30-4g)/3) = (30-4g)
    4. 6•(3/1) = 6•3 = 18
    1. For inequalities, if you multiply or divide by a negative number, you must flip the inequality symbol!
    1. So the '≥' symbol stays the same for now
    2. Our inequality now looks like this: '(30-4g) ≥ 18
    1. Remember, dealing with grouping symbols is part of the order of operations!
    1. Remember, dealing with grouping symbols is part of the order of operations!
    1. Remember, dealing with grouping symbols is part of the order of operations!
    2. Since there is no multiplication outside of the parentheses, we can just drop them, giving us '30-4g'
    1. We really began this process when we multiplied our inequality by 3 and dropped the parentheses
    1. Subtracting '30' will undo the addition of '30' on the left side
    2. On the right side: 18-30 = -12
    1. Dividing by -4 will undo the multiplication by '-4' on the left side, giving us 'g' by itself
    2. On the right: -12/-4 = 3
    3. Remember, dividing by two negative numbers produces a positive number
    1. For inequalities, if you multiply or divide by a negative number, you must flip the inequality symbol!
    1. For inequalities, if you multiply or divide by a negative number, you must flip the inequality symbol!
    2. Since we divided by -4, our '≥' symbol gets flipped to '≤'
    1. Our answer is 'g≤3'
    1. Opening up a set just means we need to draw a left curly brace, '{'
    2. Curly braces tell us we are dealing with sets
    1. So we put our inequality 'g≤3' after '{g |'
    1. Closing our set just means we need to draw a right curly brace, '}'
    2. So our answer in set-builder notation is: '{g | g≤3}'