Answer
1) The values of a that will ensure that the system has at least one solution is (a ≤ 7).
2) The values of a that will ensure that the system has no solution is (a > 5).
Step-by-step explanation
1) The question gives two inequality equations and asks for the values of a that will ensure the system has at least one solution.
x ≤ 7
x ≥ a
For this system to have at least one solution, whatever value a takes on has to make it possible to have solutions for x.
The first equation tells us that x is less than or equal to 7. So, all the feasible solutions from that expression is that x can be 7, 6, 5 and so on, basically all the numbers from 7 downwards.
So, for these solutions to continue, the second expression that x is greater than or equal to a must ensure that some of the solutions from the first equation (7 and below) must satisfy the second equation.
So, a can take on any value below 7, for this to still be possible.
a ≤ 7
Hence,
If a = 7
x ≤ 7
x ≥ 7
A solution will be x = 7
If a = 6
x ≤ 7
x ≥ 6
A solution will be x = 6 and 7
And so on, fo all the numbers from 7 below.
So, the values of a that will ensure that the system has at least one solution is (a ≤ 7).
2) This question is similar to the one above. Only that this one asks us that for which values of a does this system not have solutions.
x ≤ 5
x ≥ a
For this system to have no solution, whatever value a takes on has to make it impossible to have solutions for x.
The first equation tells us that x is less than or equal to 5. So, all the feasible solutions from that expression is that x can be 5, 4, 3 and so on, basically all the numbers from 5 downwards.
So, for this system to not have solution, the second expression that x is greater than or equal to a must ensure that none of the solutions from the first equation (5 and below) must satisfy the second equation.
Hence, as long as the value of a is greater than 5, it will be impossible to write a solution for this system.
So, a can take on any value above 5, for it to be impossible for a solution
a > 5
Hence,
If a = 6
x ≤ 5
x ≥ 6
It is impossible to write a solution that is possible for this system.
If a = 7
x ≤ 5
x ≥ 7
It is impossible to write a solution that is possible for this system.
So, the values of a that will ensure that the system has no solution is (a > 5).
Hope this Helps!!!