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Most of the time in algebra, you’ll work with equations. These are statements that involve quantities that are the same, or are supposed to be the same. But sometimes you’ll need to express the fact that quantities differ, or at least the fact that they don’t have to be the same. Such statements are called inequalities.

Not equal

When you want to indicate that two quantities are never equal, but you don’t want to specify relative size or the extent to which they’re different, you can use the “not equal to” symbol. You’ve already seen this in action. It’s an equals sign with a slash through it (≠). Here are some examples of its use:

• To state that 3 is not equal to 7/2, write 3 ≠ 7/2. • To state or require that x is never equal to 0, write x≠ 0. • To state or require that x is never equal to y, writex≠y. • To state or require that 2x is never equal to x, write 2x≠x.

Strictly larger

When a certain quantity is always larger than (or greater than) some other quantity, the “strictly larger than” symbol is used. It looks like a letter V rotated a quarter-turn counterclockwise,

or an arrowhead pointing to the right (>). When you use this symbol, remember that “larger” means “more positive” or “less negative.”

• To state that 3 is strictly larger than −7/2, write 3 > −7/2. • To state or require that x is strictly larger than 0, write x> 0. • To state or require that x is strictly larger than y, write x>y. • To state or require that 2x is strictly larger than x, write 2x>x.

Strictly smaller

When a certain quantity is always smaller than (or less than) some other quantity, the “strictly smaller than” symbol is used. It looks like a letter V rotated a quarter-turn clockwise, or an arrowhead pointing to the left (<). When you use this symbol, remember that “smaller” means “less positive” or “more negative.”

• To state that −1 is strictly smaller than 7/2, write −1< 7/2. • To state or require that x is strictly smaller than 0, write x< 0. • To state or require that x is strictly smaller than y, write x<y. • To state or require that 2x is strictly smaller than x, write 2x<x.

Larger than or equal

When a certain quantity is always larger than or equal to some other quantity, the “larger than or equal” symbol is used. It looks like a Roman numeral IV rotated a quarter-turn counter- clockwise, or an arrowhead pointing to the right with a line underneath (≥).

• To state that 3 is larger than or equal to −7/2, write 3 ≥ −7/2. • To state or require that x is larger than or equal to 0, write x≥ 0. • To state or require that x is larger than or equal to y, write x≥y. • To state or require that 2x is larger than or equal to x, write 2x≥x.

Smaller than or equal

When a certain quantity is always smaller than or equal to some other quantity, the “smaller than or equal” symbol is used. It looks like a Roman numeral VI rotated a quarter-turn clock- wise, or an arrowhead pointing to the left with a line underneath (≤).

• To state that −1 is smaller than or equal to 7/2, write −1≤ −7/2. • To state or require that x is smaller than or equal to 0, write x≤ 0. • To state or require that x is smaller than or equal to y, writex≤y. • To state or require that 2x is smaller than or equal to x, write 2x≤x.

Are you confused?

How can a quantity 2x can be strictly smaller than x, or smaller than or equal to x, as is mentioned twice in the above examples? Think for a moment about the meaning of “smaller” with respect to positive and negative numbers. Then remember what happens when you multiply a negative number by a positive

178 Equations and Inequalities

number such as 2. Once you remember this, it’s easy to see that if x is any negative number, then 2x is smaller than x.Now you can write

If x< 0, then 2x<x

and

If x≤ 0, then 2x≤x

Check these facts out with some actual numbers and you’ll see how they work. When any number is nega- tive to begin with, doubling it makes it more negative, and therefore smaller.

Logical implication

Here is a new mathematical symbol. An “if/then” statement, such as those above, can be abbreviated using a double-shafted arrow pointing to the right, often with a little extra space on either side (⇒), between the “if ” part of the statement and the “then” part. This arrow stands for the term logically implies, which in plain English translates to “means it is always true that.” (It does not mean “causes”!) With the help of this symbol, the above facts can be shortened to

(x< 0) ⇒ (2x<x) and

(x≤ 0) ⇒ (2x≤x)

Try reading these statements by saying “logically implies” or “means it is always true that” when you see the arrow.

In any logical implication of this kind, the part of the statement to the left of the arrow is called the antecedent. The part of the statement to the right of the arrow is called the consequent.

Here’s a challenge!

Write a pair of “if/then” statements that precisely define all the real numbers that, when divided by 10, become smaller than the original number.

Solution

Let’s begin by seeking out all the real numbers that become strictly smaller when we divide them by 10. It’s not difficult to see that any positive real will work. We can say that

(x> 0) ⇒ (x/10<x)

If we start with a negative real and then divide it by 10, the result gets less negative, meaning that it becomes larger. All the negative reals therefore fail to “qualify.” We want the number to get smaller, not

larger! What about 0? If we divide 0 by 10, we end up with 0 again, so 0 does not “qualify” either. We want the number to get strictly smaller! Now, addition to the above statement, we can claim its reverse:

(x/10<x)⇒ (x> 0)

This means that if we divide a real number by 10 and get a strictly smaller number, the original number

must be positive.

Logical equivalence

When a logical implication works in both directions, we have logical equivalence. This means that the left-hand part of the statement is true if and only if the right-hand part is true. The antecedent can also be the consequent, and vice-versa. To symbolize logical equiva- lence, we use a double-shafted, double-headed arrow, often with extra space on either side (⇔). We can also write the cryptic word “iff.” Now we can answer the challenge above with a single statement. We can write either

(x/10<x)⇔ (x> 0) or

(x/10<x) iff (x> 0)