We have been asked to describe the solutions of the given compound inequalities. We will do so by looking at each inequality individually, breaking each of them up as necessary. Each given inequality can be broken into two parts.
x>9 and x<9
First Inequality:&x > 9
Second Inequality:&x < 9
The first inequality tells us that we want all values greater than 9 but not 9 itself. The second inequality tells us that we want all values less than 9 but not 9 itself. Since the inequality is an "and" inequality, we need the overlapping section of these two sets. No number can be both less than and greater than 9 at the same time.
Therefore, there are no real solutions.
x<9 or x>9
First Inequality:&x < 9
Second Inequality:&x > 9
The first inequality tells us that we want all values less than 9 but not 9 itself. The second inequality tells us that we want all values greater than 9 but not 9 itself. Therefore, the solution is all real numbers below 9 or all values above 9.
x≥ 9 and x≤ 9
First Inequality:&x ≥ 9
Second Inequality:&x ≤ 9
The first inequality tells us that we want all values greater than 9 and 9 itself. The second inequality tells us that we want all values less than 9 and 9 itself. Since the inequality is an "and" inequality, we need the overlapping section of these two sets. The only value contained in the overlap is 9 itself. Hence, the only real solution is 9.
x≤ 9 or x≥ 9
First Inequality:&x ≤ 9
Second Inequality:&x ≥ 9
The first inequality tells us that we want all values less than 9 and 9 itself. The second inequality tells us that we want all values greater than 9 and 9 itself.
Therefore, the solution is all real numbers 9 and below or all real numbers 9 and above. Note that 9 is always a solution.
