4x+3<5x+4<2x+13

4x+3 < 5x+4 < 2x+13 is two separate problems.

3<x+4
-1 < x

5x+4<2x+13
3x < 9
x < 3

So, the solution is -1 < x < 3

4x+3<5x+4<2x+13

subtract 4 from each part:
4x - 1 < 5x < 2x + 9
4x-1 < 5x AND 5x < 2x+ 9
-x < 1 AND 3x < 9
x > -1 AND x < 3

or -1 < x < 3

To solve the inequality 4x + 3 < 5x + 4 < 2x + 13, we need to find the range of values that x can take while satisfying the given inequality.

First, we'll focus on the left side of the inequality, 4x + 3 < 5x + 4. To isolate x on one side, we can subtract 4x from both sides:
4x + 3 - 4x < 5x + 4 - 4x
3 < x + 4

Next, we move to the right side of the inequality, 5x + 4 < 2x + 13. To isolate x on one side, we subtract 2x from both sides:
5x + 4 - 2x < 2x + 13 - 2x
3x + 4 < 13

Now, we have two separate inequalities:
3 < x + 4 and 3x + 4 < 13

Let's solve the first inequality, 3 < x + 4:
Subtract 4 from both sides:
3 - 4 < x + 4 - 4
-1 < x

Now let's solve the second inequality, 3x + 4 < 13:
Subtract 4 from both sides:
3x + 4 - 4 < 13 - 4
3x < 9

Divide both sides by 3 to isolate x:
(3/3)x < 9/3
x < 3

Combining the results, we have:
-1 < x < 3

Therefore, the values of x that satisfy the original inequality 4x + 3 < 5x + 4 < 2x + 13 are all real numbers between -1 and 3, excluding -1 and 3.

Please note that when solving inequalities, it is important to pay attention to any possible restrictions or conditions stated in the problem.