Answer 11-2
If we have two numbers, variables, or mathematical expressions p and q, then five types
inequalities can exist:
- If p is strictly smaller than q, we write p<q.
- If p is smaller than or equal to q, we write p≤q.
- If p is not equal to q but we don’t know which is smaller, we write p≠q.
- If p is larger than or equal to q, we write p≥q.
- If p is strictly larger than q, we write p>q.
Question 11-3
Suppose we multiply an inequality through by a nonzero real number. Under what circum-
stances will the sense of the inequality be reversed? Under what circumstances will the sense
of the inequality stay the same?
Answer 11-3
The sense of the inequality is reversed if we multiply through by a negative real number, and
it remains the same if we multiply through by a positive real number.
Question 11-4
Suppose we divide an inequality through by a nonzero real number. Under what circum-
stances will the sense of the inequality be reversed? Under what circumstances will the sense
of the inequality stay the same?
Answer 11-4
The sense of the inequality is reversed if we divide through by a negative real number, and it
remains the same if we divide through by a positive real number.
Question 11-5
If a quantity x is strictly smaller than another quantity y, then we can also say x is smaller than
or equal to y. How would we write this fact entirely in logical symbols?
Answer 11-5
Remember the symbols for “strictly smaller than,” “smaller than or equal to,” and “logically
implies.” The above statement can be written symbolically as
(x<y)⇒ (x≤y)Question 11-6
We can’t square both sides of a “strictly smaller than” inequality and be sure that we’ll get
another valid statement. Provide an example that shows why.
Answer 11-6
Consider the fact that − 3 < 2. If we square both sides, we get 9 < 6, which is false. There are
infinitely many other examples.
Part Two 309