The solution of the system of inequalities `3x-7lt5+x` and `11-5xle1` on the number line is

Options

A.
B.
C.
D. None of the above

Answer

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

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

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

2. Now, divide both sides by 2:
\[ x < 6 \]

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

1. Rearranging gives:
\[ -5x \leq 1 - 11 \]
Which simplifies to:
\[ -5x \leq -10 \]

2. Now, divide both sides by -5. Remember, when dividing by a negative number, we reverse the inequality:
\[ x \geq 2 \]

### Step 3: Combine the results

From the two inequalities, we have:
1. \(x < 6\)
2. \(x \geq 2\)

Thus, the combined solution is:
\[ 2 \leq x < 6 \]

### Step 4: Represent the solution on the number line

- The solution \(x \geq 2\) is represented by a closed dot at 2 and shading to the right.
- The solution \(x < 6\) is represented by an open dot at 6 and shading to the left.

### Final Representation

On the number line, the solution can be represented as:
- A closed dot at 2 (indicating that 2 is included in the solution).
- An open dot at 6 (indicating that 6 is not included in the solution).
- The shaded region between these two points shows all the values of \(x\) that satisfy both inequalities.