Solve and graph the solution set on a number line :
`x/2 gt -1 +(3x)/4, x in N`

Answer

To solve the inequality \( \frac{x}{2} > -1 + \frac{3x}{4} \) where \( x \) is a natural number, follow these steps:

### Step 1: Write the inequality
We start with the inequality:
\[ \frac{x}{2} > -1 + \frac{3x}{4} \]

### Step 2: Move all terms involving \( x \) to one side
Subtract \( \frac{3x}{4} \) from both sides:
\[ \frac{x}{2} - \frac{3x}{4} > -1 \]

### Step 3: Find a common denominator
The common denominator for \( 2 \) and \( 4 \) is \( 4 \). Rewrite \( \frac{x}{2} \) as \( \frac{2x}{4} \):
\[ \frac{2x}{4} - \frac{3x}{4} > -1 \]

### Step 4: Combine the fractions
Now combine the fractions on the left side:
\[ \frac{2x - 3x}{4} > -1 \]
This simplifies to:
\[ \frac{-x}{4} > -1 \]

### Step 5: Eliminate the fraction
To eliminate the fraction, multiply both sides by \( -4 \). Remember to reverse the inequality sign:
\[ -x < 4 \]

### Step 6: Solve for \( x \)
Now, multiply both sides by \( -1 \) (again reversing the inequality):
\[ x > -4 \]

### Step 7: Consider the natural numbers
Since \( x \) must be a natural number, the smallest natural number is \( 1 \). Therefore, we have:
\[ x \in \{1, 2, 3, \ldots\} \]

However, since we derived \( x > -4 \), we only need to consider natural numbers less than \( 4 \):
\[ x \in \{1, 2, 3\} \]

### Step 8: Graph the solution on a number line
To graph the solution set \( \{1, 2, 3\} \) on a number line, we place closed circles at \( 1 \), \( 2 \), and \( 3 \) to indicate that these values are included in the solution set.

### Final Solution
The solution set is \( \{1, 2, 3\} \).

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