How Do You Solve a Word Problem Using a Multi-Step Inequality?

Summary

$30 was the amount she started with in her wallet
She bought four packs of gum, which can be represented as '4g', since the cost of one pack is 'g'
At least 1/3 of what was left in her wallet was spent on gas
'30-4g' represents the amount left in her wallet before buying gas
Dividing '30-4g' by '3' will give us the least amount of money she could have spent on gas, which we are told was $6
The '≥' symbol means GREATER THAN OR EQUAL TO
Dividing a negative by a negative will give you a positive
So a single pack of gum cost Nicoyia at least $3

Notes

1. We chose 'g' as our variable because 'gum' starts with a 'g'
1. Remember, 'g' represents the cost of a single pack of gum
1. Nicoyia started with $30 in her wallet
2. To find the total cost of the gum, we can multiply the 4 packs of gum by the price of a pack of gum, 'g'
1. We subtract the total spent on gum from the amount of money Nicoyia started with, $30
1. We subtract the total spent on gum from the amount of money Nicoyia started with, $30
2. 30-4•(price of a pack of gum)
1. We subtract the total spent on gum from the amount of money Nicoyia started with, $30
2. 30-4•(price of a pack of gum) = 30-4g
1. We subtract the total spent on gum, '4g', from the amount of money Nicoyia started with, '30'
2. Before buying gas, '30-4g' was the amount left in Nicoyia's wallet
3. Remember, 'g' represents the cost of a single pack of gum
1. Before buying gas, '30-4g' was the amount left in Nicoyia's wallet
2. Since she spent 1/3 of what was left in her wallet on gas, we need to divide '30-4g' by '3'
3. Realize that multiplying '30-4g' by '1/3' would be the same thing as dividing it by '3'
1. 'AT LEAST' can be represented by the '≥' symbol
1. 'AT LEAST' can be represented by the '≥' symbol
2. The '≥' symbol means GREATER THAN OR EQUAL TO
1. So this gives us the inequality: '(30-4g)/3 ≥ 6'
1. Remember, 'g' represents the cost of a single pack of gum
1. This will help us get 'g' on one side of the inequality
1. In order to undo the division by '3' on the left-hand side, we need to multiply by '3'
2. So this means we need to multiply the right-hand side by '3', as well!
1. The '3's cancel each other on the left, leaving us with (30-4g) on that side
2. On the right side, we multiply the '6' by 3, giving us '6•3', or '18'
3. (30-4g) ≥ 18
4. The '≥' symbol means GREATER THAN OR EQUAL TO
1. Since there is no multiplication outside of the parentheses, we can just drop them, giving us '30-4g'
1. Subtracting '30' will undo the addition of '30', giving us '-4g' on the left
2. On the right side: 18-30 = -12
1. Subtracting '30' will undo the addition of '30', giving us '-4g' on the left
2. On the right side: 18-30 = -12
3. -4g ≥ -12
4. The '≥' symbol means GREATER THAN OR EQUAL TO
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 a negative by a negative gives you a positive
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 a negative by a negative gives you a positive
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. Now that 'g' is by itself on one side, we have solved for 'g'!
1. Our answer is 'g≤3', so a single pack of gum must be $3 or less

Nicoyia has $30 in her wallet. She went to the grocery store and bought 4 packs of gum. She looked at how much money she had left and realized she should spend at least one third of what was left on gas. If she spent $6 on gas, how much was the cost of each pack of gum?

Summary

Notes