Let's solve this problem step-by-step to find the necessary values and understand the inequality that represents the number of calories consumed.
1. Number of calories in [tex]\(\frac{3}{4}\)[/tex] cups of red seedless grapes:
We know that [tex]\(\frac{1}{4}\)[/tex] cup of red seedless grapes contains 30 calories.
To find the calories in [tex]\(\frac{3}{4}\)[/tex] cups, we multiply the calories in [tex]\(\frac{1}{4}\)[/tex] cup by the number of [tex]\(\frac{1}{4}\)[/tex] cups in [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\text{Calories in } \frac{3}{4} \text{ cups} = 30 \times 3 = 90 \text{ calories}
\][/tex]
2. Number of calories in [tex]\(1 \frac{1}{4}\)[/tex] cups of red seedless grapes:
[tex]\(1 \frac{1}{4}\)[/tex] cups can be rewritten as [tex]\(1 + \frac{1}{4}\)[/tex] cups. Since 1 cup contains 4 quarters and each quarter cup contains 30 calories:
[tex]\[
\text{Calories in } 1 \frac{1}{4} \text{ cups} = \text{Calories in 1 cup} + \text{Calories in } \frac{1}{4} \text{ cup}
\][/tex]
Now, calculate the calories separately:
[tex]\[
\text{Calories in 1 cup} = 30 \times 4 = 120 \text{ calories}
\][/tex]
[tex]\[
\text{Calories in } \frac{1}{4} \text{ cup} = 30 \text{ calories}
\][/tex]
Adding them together:
[tex]\[
\text{Calories in } 1 \frac{1}{4} \text{ cups} = 120 + 30 = 150 \text{ calories}
\][/tex]
3. The inequality representing the calories when eating more than [tex]\(\frac{3}{4}\)[/tex] but less than [tex]\(1 \frac{1}{4}\)[/tex] cups:
The range of cup measurements is between [tex]\(\frac{3}{4}\)[/tex] and [tex]\(1 \frac{1}{4}\)[/tex] cups. In calorie terms, this translates to more than 90 calories but less than 150 calories.
Therefore, the inequality can be expressed as:
[tex]\[
90 \text{ calories} < \text{Calories consumed} < 150 \text{ calories}
\][/tex]
In conclusion:
- The number of calories in [tex]\(\frac{3}{4}\)[/tex] cups of red seedless grapes is [tex]\(\boxed{90}\)[/tex].
- The number of calories in [tex]\(1 \frac{1}{4}\)[/tex] cups of red seedless grapes is [tex]\(\boxed{150}\)[/tex].
- The inequality representing the number of calories when eating between [tex]\(\frac{3}{4}\)[/tex] and [tex]\(1 \frac{1}{4}\)[/tex] cups of grapes is [tex]\(\boxed{90 < \text{Calories consumed} < 150}\)[/tex].
