a hair salon sells a certain brand of shampoo in 20 ounces and 50 ounces of cost one one shampoo in 20 ounces is 20 dollars and 50 ounces is 42 dollars if total of 90 bottles were sold between 2900 dollars and 3120 dollars.which could be no of 50 ounces bottle sold

Let's solve this problem step by step.

### Given:
- Number of bottles sold = 90
- Total revenue between $2900 and $3120
- Price of a 20-ounce bottle = $20
- Price of a 50-ounce bottle = $42

### Variables:
- Let $ x $ be the number of 50-ounce bottles sold.
- Let $ y $ be the number of 20-ounce bottles sold.

### Equations:
1. $ x + y = 90 $  (Total number of bottles sold)
2. $ 42x + 20y $ is the total revenue, which lies between $2900 and $3120.

### Substitute $ y = 90 - x $ in the revenue equation:
\[
42x + 20(90 - x) \text{ should be between 2900 and 3120}
\]

Simplify the revenue equation:
\[
42x + 1800 - 20x = 22x + 1800
\]
\[
2900 \leq 22x + 1800 \leq 3120
\]

Subtract 1800 from all sides:
\[
1100 \leq 22x \leq 1320
\]

Now, divide by 22:
\[
50 \leq x \leq 60
\]

### Conclusion:
The number of 50-ounce bottles sold could be any integer between 50 and 60.

Do you want details or have any questions?  
Here are 8 related questions:

1. How do we find the number of 20-ounce bottles sold?
2. What would be the exact total revenue if 55 bottles of 50 ounces were sold?
3. How would the problem change if the 20-ounce bottle was sold for $25 instead?
4. What is the range of possible 20-ounce bottles sold given the current conditions?
5. Could the total revenue have been exactly $3000? If so, how many 50-ounce bottles would have been sold?
6. How does the revenue equation change if the total number of bottles sold is different?
7. What happens to the solution if the prices of the bottles increase by 10%?
8. How do you verify if a particular number of bottles sold satisfies all conditions?

**Tip:** When dealing with inequalities, always remember to adjust all parts of the inequality when performing arithmetic operations.

Learn how to solve a linear equation problem involving the sales of shampoo bottles in different sizes (20 ounces and 50 ounces). This problem covers concepts of linear equations and inequalities. Suitable for students in Grades 7-9.

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