For to change the signs of all the terms is equivalent to subtracting each member from 0=0.

IV. If two or more inequalities subsisting in the same sense, be added, member to member, the resulting inequality will subsist in the same sense as the given inequalities.

For if

a> b, a' > b', a" > b", .....,

then from (196),

a-b, a'-b', a"-b",

.....

are all positive; and the sum of these quantities,

a—b+a'—b'+a”—b”, or (a+a'+a′′)—(b+b'+b′′),

is therefore positive. Hence,

a+a+a">b+b'+b".

It is evident that if one inequality be subtracted from another established in the same sense, the result will not always be an inequality subsisting in the same sense. Thus, it is evident that we may have

a> b and a'>b',

in which a―a' may be greater than b—b', less than b—b', or equal to b-b'.

V. If one inequality be subtracted from another established in a contrary sense, the result will be an inequality established in the same sense as the minuend.

and

For, if

ab a' <b',

(1)

(2)

then ab is positive and a'-b' is negative; therefore, a-b— (a'-'), or its equal (a-a')-(b-b') must be positive, and we

shall have

a-a'>b-b',

an inequality subsisting in the same sense as (1).

If (1) be subtracted from (2), member from member, it can be shown, in like manner, that

a'—a < b'—b.

VI. An inequality will still subsist in the same sense, if both members be multiplied or divided by the same positive quantity.

1

m

For suppose m to be essentially positive, and

a> b.

Then since a-b is positive, we shall have both m(a—b) and (a-b) positive. Therefore,

VII. If both members of an inequality be multiplied or divided by the same negative quantity, the sign of inequality will be reversed. For, to multiply or divide by a negative quantity will change the signs of all the terms, and consequently reverse the sign of inequality, (III).

VIII. If two inequalities subsisting in the same sense be multiplied together, member by member, the sign of inequality remains the same when more than two of the members are positive, but is reversed when more than two of the members are negative.

Products, aa'> bb'

aa' <bb' —aa' < bb' aa'>-vv'

The first two results are evident from the fact that when the two members of an inequality are both positive, the greater member has the greatest numerical value; but when the two members are both negative, the greater memor has the least numerical value.

The other two results are evident from the fact that any positive quantity is greater than any negative quantity.

It will be found that if two of the four members are positive and two negative, the result will be indefinite.

REDUCTION OF INEQUALITIES.

200. The Reduction of an inequality consists in transforming it in such a manner that one member shall be the unknown quantity standing alone, and the other member a known expression. The inequality will then denote one limit of the unknown quantity.

201. The principles just established may now be applied in the reduction of inequalities of the first degree.

Thus, let it be required to find the limit of in the inequality,

202. If there be given an inequality and an equation, containing two unknown quantities, the limit of each unknown quantity may be found, by a process of elimination.

1. Given 2x+5y > 16 and 2x+y= 12, to find the limits of x

and y.

If we subtract the equation from the inequality, the result will

be an inequality subsisting in the same sense, (199, I, 2), and x will be eliminated.

From

subtract

Thus,

If we substitute 1 for y in the equation, the first member will be made less than the second; and we shall have

The limit of x may be found in a different manner, as follows: From equation (2),

y = 12-2x.

Substituting this value of y in (1), we have

whence,

or,

2x+60-10x16,

-8x-44,
x < 51.

Thus we may eliminate between equalities and inequalities, either by addition and subtraction, or by substitution. Let it be remem

bered, however, that when an inequality is subtracted from an equa tion, the sign of inequality will be reversed; (199, II).

EXAMPLES FOR PRACTICE.

1. Given 2x+4y> 30 and 3x+2y= 31, to find the limits of and y. Ans. x <8; y > 31.

2. Given 4x-3y < 15 and 8x+2y=46, to find the limits of x Ans. x < 51; y > 2.

and y. 3. Given 7x-10y <59 and 4x+5y= 68, to find the limits of x Ans. x < 13; y > 31.

and y.

SECTION III.

POWERS AND ROOTS.

INVOLUTION.

203. A Power of a quantity is the product obtained by taking the quantity some number of times as a factor; the quantity is then said to be raised, or involved.

204. Involution is the process of raising a quantity to any given power.

205. Involution is always indicated by an exponent, which expresses the name of the power, and shows how many times the quantity is taken as a factor.

Thus, let a represent any quantity whatever; then,

206. The Square of a quantity is its second

The Cube of a quantity is its third power.

power; and

207. A Perfect Power is a quantity that can be exactly produced by taking some other quantity a certain number of times as a factor. Thus, x2-2xy+y' is a perfect power, because it is equal to (x-y) (x-y).

POWERS OF MONOMIALS.

208. A simple factor may be raised to any power by giving it an exponent which expresses the name or degree of the required power. And if a quantity consists of two or more factors, it is evident that as often as the quantity is repeated, each factor will be repeated. Thus,

(ab)2=abab=aaɣbb = a2b3.