Solve `15-2(2x-1) lt 15, x in Z`

Answer

To solve the inequality \( 15 - 2(2x - 1) < 15 \) where \( x \) belongs to the integers (denoted as \( x \in \mathbb{Z} \)), we will follow these steps:

### Step 1: Simplify the left-hand side (LHS)
Start by distributing the \(-2\) across the expression in the parentheses:
\[ 15 - 2(2x - 1) < 15 \]
This becomes:
\[ 15 - 4x + 2 < 15 \]

### Step 2: Combine like terms
Now, combine the constant terms on the left-hand side:
\[ (15 + 2) - 4x < 15 \]
This simplifies to:
\[ 17 - 4x < 15 \]

### Step 3: Isolate the variable term
Next, we need to isolate the term involving \( x \). Subtract 17 from both sides:
\[ -4x < 15 - 17 \]
This simplifies to:
\[ -4x < -2 \]

### Step 4: Divide by -4
Now, divide both sides by \(-4\). Remember, when you divide or multiply an inequality by a negative number, you must reverse the inequality sign:
\[ x > \frac{-2}{-4} \]
This simplifies to:
\[ x > \frac{1}{2} \]

### Step 5: Determine integer solutions
Since \( x \) must be an integer, we look for integer values greater than \( \frac{1}{2} \). The smallest integer greater than \( \frac{1}{2} \) is \( 1 \). Therefore, the integer solutions are:
\[ x = 1, 2, 3, 4, \ldots \]

### Final Solution
The solution set for the inequality \( 15 - 2(2x - 1) < 15 \) where \( x \in \mathbb{Z} \) is:
\[ x \in \{ 1, 2, 3, 4, \ldots \} \]