If `x + y gt 5` and `x - y gt 3`, then which of the following inequalities gives all the possible values of x ?

Options

A. `x gt 3`
B. `x gt 4`
C. `x gt 5`
D. `x lt 5`

Answer

To solve the problem, we need to analyze the given inequalities step by step.

**Step 1: Write down the inequalities**
We have two inequalities:
1. \( x + y > 5 \) (Inequality 1)
2. \( x - y > 3 \) (Inequality 2)

**Step 2: Add the two inequalities**
We will add both inequalities together:
\[ (x + y) + (x - y) > 5 + 3 \]
This simplifies to:
\[ x + y + x - y > 8 \]
The \(y\) terms cancel out:
\[ 2x > 8 \]

**Step 3: Solve for x**
Now, we divide both sides of the inequality by 2:
\[ x > 4 \]

**Conclusion:**
The solution shows that \(x\) must be greater than 4. Therefore, the inequality that gives all possible values of \(x\) is:
\[ x > 4 \]

**Step 4: Identify the correct option**
Looking at the options provided:
- Option A: \(x > 3\)
- Option B: \(x > 4\) (This is our answer)
- Option C: \(x > 5\)
- Option D: \(x < 5\)

The correct answer is Option B: \(x > 4\).

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