Solve and present on graph 3<2x+3<=9+x ?

To solve the inequality 3 < 2x + 3 ≤ 9 + x, we need to isolate x.

First, let's subtract 3 from all parts of the inequality:

3 - 3 < 2x + 3 - 3 ≤ 9 + x - 3

0 < 2x ≤ 6 + x

Next, let's subtract x from all parts of the inequality:

0 - x < 2x - x ≤ 6 + x - x

-x < x ≤ 6

Now, let's split the inequality into two separate inequalities:

-x < x and x ≤ 6

For the first inequality, let's add x to both sides:

-x + x < x + x

0 < 2x

Divide both sides by 2:

0/2 < 2x/2

0 < x

For the second inequality, x ≤ 6, we don't need to make any changes.

So, the solution to the inequality 3 < 2x + 3 ≤ 9 + x is 0 < x and x ≤ 6.

Now, let's present this solution on a graph:

On the x-axis, mark the numbers from 0 to 6. Since the inequality includes 0 and 6, we will use closed circles at these points.

Then, shade the region between 0 and 6, including 0 but not including 6, since the inequality is x ≤ 6.

Finally, draw an arrow to the right of 6 to represent the solution 0 < x.

The graph should look like this:

```
|-------------------|-------------------|
0 6
```

The shaded region represents the solution to the inequality 3 < 2x + 3 ≤ 9 + x.