Final answer:
After analyzing the given options, we can conclude that the compound inequalities with no solution are options a and c, with option a ('x ≤ -2' and '2x > 6') being the clearest one due to its proper notation.
Step-by-step explanation:
The given question asks us to identify which compound inequality has no solution. To find out, we must analyze each pair of inequalities to determine if there is any overlap in the solutions or if they contradict each other, meaning there is no set of values for 'x' that could satisfy both inequalities at the same time.
- Option a: 'x ≤ -2' and '2x > 6' can be simplified to 'x ≤ -2' and 'x > 3'. There is no overlap since no number can be both less than or equal to -2 and greater than 3 at the same time, so this has no solution.
- Option b: 'x ≤ -1' and '5x ≤ 5' can be simplified to 'x ≤ -1' and 'x ≤ 1'. There is an overlap in the solutions (all numbers less than or equal to -1), so this has a solution.
- Option c: 'x ≤ -1' and '3x2 ≤ -3' is incorrect notation but assuming '3x^2 ≤ -3' which is impossible because a squared term multiplied by a positive number can't be negative. This has no solution.
- Option d: 'x ≤ -2' and '4x ≤ -8' can be simplified to 'x ≤ -2' and 'x ≤ -2'. These are the same inequalities, so they have the same set of solutions, in this case, all x less than or equal to -2.
Thus, the compound inequalities with no solution are options a and c. However, since option c has improper notation, the clearest answer is option a, 'x ≤ -2' and '2x > 6'.