Final answer:
To determine which compound inequality has no solution, we evaluate each option by solving the inequalities separately and checking for common values of x. Option A and Option D have no common values of x, so the compound inequalities have no solution.
Step-by-step explanation:
To determine which compound inequality has no solution, we need to analyze each option and see if there are any values of x that make both inequalities true. Let's evaluate each option:
A. xs-2 and 2x > 6
B. xs-1 and 5x < 5
C. xs-1 and 3x²-3
D. xs-2 and 4x < -8
By solving each inequality, we can determine if there are any common values for x. If there are no common values, then the compound inequality has no solution.
A. For xs-2 and 2x > 6, solving both inequalities separately:
xs-2: x > 2
2x > 6: x > 3
There is no common value for x that satisfies both inequalities, so the compound inequality has no solution.
Using the same approach, we can evaluate options B, C, and D:
B. For xs-1 and 5x < 5, solving both inequalities separately:
xs-1: x > 1
5x < 5: x < 1
Again, there is no common value for x that satisfies both inequalities, so the compound inequality has no solution.
C. For xs-1 and 3x²-3, solving both inequalities separately:
xs-1: x > 1
3x²-3: x² > 1
The second inequality can be further simplified by subtracting 1 from both sides: x²-1 > 0. This represents a parabola with roots at x = -1 and x = 1. Since the inequality is x > 1, the only common value with xs-1 is x = 1. Thus, this compound inequality has a solution.
D. For xs-2 and 4x < -8, solving both inequalities separately:
xs-2: x > 2
4x < -8: x < -2
Again, there is no common value for x that satisfies both inequalities, so the compound inequality has no solution.