The least positive integer `x` satisfying `|x+1|+|x-4|gt7` is

Options

A. `x=5`
B. `x=6`
C. `x=7`
D. `x=8`

Answer

To solve the inequality \( |x + 1| + |x - 4| > 7 \), we will analyze the expression by considering the critical points where the expressions inside the absolute values change their signs. The critical points here are \( x = -1 \) and \( x = 4 \).

### Step 1: Identify the intervals
The critical points divide the number line into three intervals:
1. \( x < -1 \)
2. \( -1 \leq x < 4 \)
3. \( x \geq 4 \)

### Step 2: Analyze each interval

**Interval 1: \( x < -1 \)**
- In this interval, both \( x + 1 \) and \( x - 4 \) are negative.
- Therefore, \( |x + 1| = -(x + 1) = -x - 1 \) and \( |x - 4| = -(x - 4) = -x + 4 \).
- The inequality becomes:
\[ -x - 1 - x + 4 > 7 \]
Simplifying this:
\[ -2x + 3 > 7 \implies -2x > 4 \implies x < -2 \]
- Since \( x < -1 \) is already our interval, this portion gives us \( x < -2 \).

**Interval 2: \( -1 \leq x < 4 \)**
- Here, \( x + 1 \) is non-negative and \( x - 4 \) is negative.
- Therefore, \( |x + 1| = x + 1 \) and \( |x - 4| = -(x - 4) = -x + 4 \).
- The inequality becomes:
\[ x + 1 - x + 4 > 7 \]
Simplifying this:
\[ 5 > 7 \]
- This is false, so there are no solutions in this interval.

**Interval 3: \( x \geq 4 \)**
- In this interval, both \( x + 1 \) and \( x - 4 \) are non-negative.
- Therefore, \( |x + 1| = x + 1 \) and \( |x - 4| = x - 4 \).
- The inequality becomes:
\[ x + 1 + x - 4 > 7 \]
Simplifying this:
\[ 2x - 3 > 7 \implies 2x > 10 \implies x > 5 \]

### Step 3: Combine the results
From our analysis:
- From Interval 1: \( x < -2 \) (not a positive integer)
- From Interval 2: No solutions
- From Interval 3: \( x > 5 \)

### Step 4: Find the least positive integer
The least positive integer satisfying \( x > 5 \) is \( x = 6 \).

### Final Answer
The least positive integer \( x \) satisfying \( |x + 1| + |x - 4| > 7 \) is \( \boxed{6} \).