Final answer:
The system of inequalities has no solutions when the value of 'a' is less than or equal to 2, because for any x greater than 1, no value for 'a' can satisfy both inequalities simultaneously.
Step-by-step explanation:
We are looking for the value of a for which the system of inequalities 3−7x < 3x−7 and 1+2x < a+x has no solutions. First, let's simplify each inequality separately:
- For the first inequality, 3−7x < 3x−7, we can combine like terms by adding 7x to both sides and adding 7 to both sides to get 10 < 10x, which simplifies to 1 < x.
- For the second inequality, 1+2x < a+x, we subtract x from both sides to get 1+x < a.
The system will have no solutions if there is no overlap in the x-values that satisfy both inequalities. From the first inequality, x must be greater than 1. For the second inequality to have no values that overlap with this condition, a must be less than or equal to 2, since x would need to be less than 1 to satisfy 1+x < a. Thus, the system 3−7x < 3x−7, 1+2x < a+x has no solutions when a ≤ 2.