Solve the system of inequalities : `3x-7<5+x`, `11-5xlt=1` and represent the solutions on the number line.

Explanation

To solve the system of inequalities \(3x - 7 < 5 + x\) and \(11 - 5x \leq 1\), we will solve each inequality step by step.

### Step 1: Solve the first inequality \(3x - 7 < 5 + x\)

1. **Rearranging the inequality**:
\[ 3x - x < 5 + 7 \]
This simplifies to:
\[ 2x < 12 \]

2. **Dividing by 2**:
\[ x < 6 \]

### Step 2: Solve the second inequality \(11 - 5x \leq 1\)

1. **Rearranging the inequality**:
\[ 11 - 1 \leq 5x \]
This simplifies to:
\[ 10 \leq 5x \]

2. **Dividing by 5**:
\[ 2 \leq x \]
or equivalently,
\[ x \geq 2 \]

### Step 3: Combine the results from both inequalities

From the first inequality, we have:
\[ x < 6 \]
From the second inequality, we have:
\[ x \geq 2 \]

### Step 4: Write the combined solution

The combined solution is:
\[ 2 \leq x < 6 \]

### Step 5: Represent the solution on the number line

To represent this on a number line:
- Draw a number line.
- Shade the region starting from 2 (including 2, so we use a closed circle) to 6 (not including 6, so we use an open circle).

### Final Answer

The solution set is \(x \in [2, 6)\).

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