If `7x + 2 ge x -2 and 7+ 2x ge 3 + 3x:` then x can take which of the following values ?

Options

A. 5
B. 3
C. `-3`
D. `-5`

Answer

To solve the inequalities \( 7x + 2 \geq x - 2 \) and \( 7 + 2x \geq 3 + 3x \), we will break it down step by step.

### Step 1: Solve the first inequality \( 7x + 2 \geq x - 2 \)

1. Start by isolating \( x \):
\[ 7x + 2 \geq x - 2 \]
Subtract \( x \) from both sides:
\[ 7x - x + 2 \geq -2 \]
This simplifies to:
\[ 6x + 2 \geq -2 \]

2. Next, subtract 2 from both sides:
\[ 6x \geq -2 - 2 \]
This simplifies to:
\[ 6x \geq -4 \]

3. Finally, divide both sides by 6:
\[ x \geq -\frac{4}{6} \]
Simplifying gives:
\[ x \geq -\frac{2}{3} \]

### Step 2: Solve the second inequality \( 7 + 2x \geq 3 + 3x \)

1. Start by isolating \( x \):
\[ 7 + 2x \geq 3 + 3x \]
Subtract \( 2x \) from both sides:
\[ 7 \geq 3 + 3x - 2x \]
This simplifies to:
\[ 7 \geq 3 + x \]

2. Next, subtract 3 from both sides:
\[ 7 - 3 \geq x \]
This simplifies to:
\[ 4 \geq x \]
or
\[ x \leq 4 \]

### Step 3: Combine the results

From the two inequalities, we have:
1. \( x \geq -\frac{2}{3} \)
2. \( x \leq 4 \)

Thus, the combined solution is:
\[ -\frac{2}{3} \leq x \leq 4 \]

### Step 4: Determine which values from the options fit this range

The options given are \( 5, 3, -3, -5 \):
- \( 5 \) is not in the range.
- \( 3 \) is in the range.
- \( -3 \) is not in the range.
- \( -5 \) is not in the range.

### Final Answer:
The value that \( x \) can take from the options is \( 3 \).

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