1

–

2

1

–

2

inequality 267

To examine this argument, note that it is indeed the

case that P(1) is valid, and it is true that P(k) implies

P(k+1)—for almost all values of k. The fault in the

argument above is that although P(2) establishes P(3),

and P(3) establishes P(4),and so forth, it is not the case

that P(1) establishes P(2):removing one horse from a

field containing just two horses does not, alas, establish

that both horses are the same color. In our chain of

dominoes, all but the first two dominoes are properly

spaced: the first domino does not topple the second

domino during its fall, and the chain remains standing.

This illuminates that one must always be careful that

any argument presented in a proof by induction is

indeed valid for all values of k.

See also

PROOF

;

SUMS OF POWERS

.

inductive reasoning See

DEDUCTIVE

/

INDUCTIVE

REASONING

.

inequality A mathematical statement that one quan-

tity or expression is greater than or less than another is

called an inequality. The following symbols are used:

a> b, a is greater than b

a< b, a is less than b

a≥b, a is greater than or equal to b

a≤b, a is less than or equal to b

Inequalities satisfy a number of

ORDER PROPERTIES

.

An inequality is called closed or unconditional if it

holds for all values of the variables, if any, that appear

in the equation. For instance, the inequalities 3 ≤5,

x2+1 > x2, and 5 + y2> 4yare closed inequalities for

they are always true. Inequalities that are not closed are

called open or conditional. The set of values of the vari-

ables that appear in the inequality that make the state-

ment true is called the solution set of the inequality. For

example, the open inequality 2x+ 1 > 7 has as a solu-

tion set the set of all real numbers xfor which x> 3.

The solution set of the open inequality a2+ b2< 0 is the

EMPTY SET

.

Open inequalities can be solved in much the same

manner as equations. As the order properties show, one

can add or subtract the same quantity to both sides of

the inequality and preserve the inequality, or can multi-

ply the inequality through by a positive quantity. Multi-

plying through by a negative quantity changes the sense

of the inequality. (For example, if a< b, then subtracting

the quantity a+ bfrom both sides yields –b< –a. This

shows that the effect of multiplying through by –1 is to

reverse the sense of the inequality.) For example, one can

solve the inequality 2x+ 1 > 7, as follows:

2x+ 1 > 7

2x> 6

×2x> ×6

x> 3

indeed yielding the solution set {x:x > 3}.

It is worth noting that if a· b> 0, then we can be

sure that aand bare either both positive or both nega-

tive. If, on the other hand, a· b< 0, then we can be

sure that aand bhave opposite signs. These observa-

tions are essential for solving inequalities involving a

single variable raised to the second power. For exam-

ple, to solve the open inequality x2+ x– 2 > 0, factor

the

QUADRATIC

to obtain (x+ 1)(x– 2) > 0 and exam-

ine the two possible scenarios. Either x+ 1 and x– 2

are both greater than zero (yielding that xmust be

greater than 2), or x+ 1 and x– 2 are both less than

zero (yielding that xmust be less than –1). Thus the

solution set to the inequality is the set of all real num-

bers xwith x> 2 or x< –1.

A single inequality in two variables defines a region

in the plane. For example, the inequality 2x+ y≥3 is

satisfied by the points (0,3), (5,0), and (2,2), for

instance. In this example, the complete solution set is

the closed

HALF

-

PLANE

sitting above the line y= –2x+

3 in the plane.

There are a number of standard inequalities in

mathematics:

Triangle inequality: For any triangle with side-lengths

a,b, and cwe have: a+ b> c.

Arithmetic-geometric mean inequality: For nonnegative

numbers a1,a2,…,anwe have:

Bernoulli’s inequality: For any real number xgreater

than 1 and positive integer n:

(1 + x)n> 1 + nx