which of these are equivalent?
I. x>-3 II. X-4<-8 III. -2x<8
a. I only
b. I and III only
c. I, II, and III
d.none of them
To determine which of the given inequalities are equivalent, we need to solve each inequality to identify the range of values that satisfy it.
I. x > -3:
To solve this inequality, we need to isolate the variable x. Since there are no operations applied to x, we can directly state that x can take any value greater than -3.
II. x - 4 < -8:
To solve this inequality, we'll add 4 to both sides to isolate the variable x. This gives us: x < -4. We can interpret this inequality as x being any value less than -4.
III. -2x < 8:
To solve this inequality, we'll divide both sides by -2. However, since we're dividing by a negative number, we need to flip the inequality sign. Thus, we have x > -4. This inequality implies that x can take any value greater than -4.
Comparing the results of each inequality:
I. x > -3
II. x < -4
III. x > -4
Both the first and third inequalities (I and III) state that x can be any value greater than -3 or -4. Therefore, the equivalent inequalities are I and III.
So, the correct answer is b. I and III only