Answer: The first option:
x ≤ -2 and 2x ≥ 6
Explanation:
A compound inequality will have no solution if the inequalities are contradictory.
An example of this would be:
x > 3 and x < 1
There is no value of x that is at the same time larger than 3, and smaller than 1.
Now, let's analyze the options.
1) x ≤ -2 and 2x ≥ 6
This has no solution, because if x ≥ -2, the maximum value that x can take is x = -2
Replacing that in the other inequality we get:
2*(-2) > 6
-4 > 6
This is false, then this compound inequality has no solution.
2) x ≤ -1 and 5*x < 5
This ineqalty has infinite solutions, one can be x = -2
-2 ≤ -1 and 5*(-2) < 5
are both true.
3) x ≤ -1 and 3x ≥ -3
A solution for this can be x = -1
-1 ≤ -1 is true
3*(-1) ≥ -3
-3 ≥ -3 is true.
Then we have at least one solution here.
4) x ≤ -2 and 4x ≤ -8
Here we have infinite solutions, one can be x = -10
-10 ≤ - 2 is true
4*(-10) ≤ -8
-40 ≤ - 8 is also true.
Then the only option that has no solutions is the first one.