Final answer:
The inequality ax + b < ax + c simplifies to b < c, which is always true regardless of x. For ax + b > ax + c to be true, b would have to be greater than c, which contradicts the original condition. Therefore, there are no real values of x that satisfy ax + b > ax + c.
Step-by-step explanation:
If we have the inequality ax + b < ax + c, to find out if this inequality is true for all real values of x, we should investigate the relationship between b and c.
Since the terms involving x on both sides of the inequality are identical, they can be subtracted from both sides to simplify the inequality to b < c.
This indicates that no matter what value of x we choose, the inequality will always depend on the relationship between b and c.
Conversely, when considering ax + b > ax + c, and we apply the same logic, subtracting ax from both sides yields b > c. So, for the inequality ax + b > ax + c to be true, b must be greater than c.
However, since we know that b < c is true for all real values from the original inequality, there are no real values of x that will make ax + b > ax + c true. Therefore, the solution set for the inequality is empty.