Solve the inequality `5-3xlt17,x inZ`. Represent the solution set on the number line.

Answer

To solve the inequality \( 5 - 3x < 17 \) where \( x \) is an integer, we can follow these steps:

### Step 1: Isolate the term with \( x \)
Start by moving the constant term on the left side to the right side of the inequality.

\[ 5 - 3x < 17 \]

Subtract 5 from both sides:

\[ -3x < 17 - 5 \]

\[ -3x < 12 \]

### Step 2: Divide by -3
Now, we need to divide both sides by -3. Remember that when we divide or multiply both sides of an inequality by a negative number, we must reverse the inequality sign.

\[ x > \frac{12}{-3} \]

\[ x > -4 \]

### Step 3: Consider the integer constraint
Since \( x \) must be an integer, we can express the solution set as:

\[ x \in \{ -3, -2, -1, 0, 1, 2, 3, \ldots \} \]

### Step 4: Represent the solution set on the number line
To represent this on a number line, we will mark the point -4 and indicate that \( x \) can take any integer value greater than -4. We will use an open circle at -4 to show that -4 is not included in the solution set.

The number line representation will look like this:

```
<----|----|----|----|----|----|----|----|----|---->
-5 -4 -3 -2 -1 0 1 2 3 4
(------------------->
```

### Final Solution
The solution to the inequality \( 5 - 3x < 17 \) is \( x > -4 \) where \( x \) is an integer, represented on the number line as shown above.

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