Solve the following systems of inequalities for all `xinR`
 In the first four papers each of 100 marks, Rishi got 95, 72, 73, 83 marks. If he wants an average of greater than or equal to 75 marks and less than 80 marks find the range of marks he should score in fifth paper.

Answer

To solve the problem step by step, we need to determine the range of marks Rishi should score in his fifth paper to achieve an average between 75 and 80 marks.

### Step 1: Calculate the total marks obtained in the first four papers
Rishi's scores in the first four papers are:
- Paper 1: 95
- Paper 2: 72
- Paper 3: 73
- Paper 4: 83

**Total marks from the first four papers:**
\[ 95 + 72 + 73 + 83 = 323 \]

### Step 2: Set up the inequality for the average
Let \( x \) be the marks Rishi scores in the fifth paper. The average of the five papers should be greater than or equal to 75 and less than 80. Therefore, we can write the inequality as:
\[ 75 \leq \frac{323 + x}{5} < 80 \]

### Step 3: Multiply the entire inequality by 5
To eliminate the fraction, we multiply all parts of the inequality by 5:
\[ 75 \times 5 \leq 323 + x < 80 \times 5 \]
This simplifies to:
\[ 375 \leq 323 + x < 400 \]

### Step 4: Solve the inequalities
Now, we will solve each part of the inequality separately.

1. **For the left part of the inequality:**
\[ 375 \leq 323 + x \]
Subtract 323 from both sides:
\[ 375 - 323 \leq x \]
\[ 52 \leq x \]
So, we have:
\[ x \geq 52 \]

2. **For the right part of the inequality:**
\[ 323 + x < 400 \]
Subtract 323 from both sides:
\[ x < 400 - 323 \]
\[ x < 77 \]

### Step 5: Combine the results
Combining the results from both parts, we have:
\[ 52 \leq x < 77 \]

### Final Answer
The range of marks Rishi should score in the fifth paper is:
\[ x \in [52, 77) \]