4x + 3 /leq 5x - 3 or 1 - 2x < -3x + 3

To solve the inequalities \(4x + 3 \leq 5x - 3\) and \(1 - 2x < -3x + 3\), we can solve them one at a time.

### Inequality 1: \(4x + 3 \leq 5x - 3\)

1. Subtract \(4x\) from both sides:
\[
3 \leq 5x - 4x - 3
\]
This simplifies to:
\[
3 \leq x - 3
\]

2. Add \(3\) to both sides:
\[
3 + 3 \leq x
\]
Thus, we have:
\[
6 \leq x \quad \text{or} \quad x \geq 6
\]

### Inequality 2: \(1 - 2x < -3x + 3\)

1. Add \(2x\) to both sides:
\[
1 < -3x + 2x + 3
\]
This simplifies to:
\[
1 < -x + 3
\]

2. Subtract \(3\) from both sides:
\[
1 - 3 < -x
\]
Which simplifies to:
\[
-2 < -x
\]

3. Multiply both sides by \(-1\) (remember, this reverses the inequality):
\[
2 > x \quad \text{or} \quad x < 2
\]

### Combining the Solutions

We have two inequalities from our solutions:

1. \(x \geq 6\)
2. \(x < 2\)

Since there are no values of \(x\) that satisfy both inequalities simultaneously, the solution set is:

\[
\text{No solution}
\]

In conclusion, there is no value of \(x\) that satisfies either of the inequalities.