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

and y.

= 12, to find the limits of x

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 = 12-2.x.

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

2x+60-10x> 16,

whence, or,

-8x-44,
x < 51.

Thus we may eliminate between equalities and inequalities, either by addition and subtraction, or by substitution. Let it be remembered, however, that when an inequality is subtracted from an equa tion, the sign of inequality will be reversed; (199, II).

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

and

y.

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

3. Given 7x-10y<59 and 4x+5y

and y.

Ans. x <51; y > 2.

= 68, to find the limits of Ans. x <13; y>3}.

4. Given 5x+3y > 121 and 7x+4y= 168, to find the limits of

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 power; and

The Cube of a quantity is its third power.

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, x-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 = aabb = a2b3.

And in general, if abc.....k represent the product of any number of factors, and n any exponent, we shall have

(abc.....k)"a"b"c".....k".

That is,

The nth power of the product of two or more factors is equal to the product of the nth powers of those factors.

209. If it be required to involve a quantity which is already a power, the exponent of the quantity will be taken as many times as there are units in the exponent of the required power. Thus,

And in general, am raised to the nth power will be

(am)n = amn.

That is, the

If the mth power of a quantity be raised to the nth power, result will be a power of the quantity expressed by the product of m and n.

210. With respect to signs, it is obvious that if a positive quantity be involved to any power whatever, the result will be positive.

But if a negative quantity be involved, the successive powers will be alternately positive and negative; for, it has been shown that the product of an even number of negative factors is positive, and the product of an odd number of negative factors is negative, (67).

To deduce this law of signs in an experimental way, let it be required to involve a to successive powers. By the principles of multiplication, we shall have,

-a

(-a)'= (-a)X(-a) = +a";
(—a)3

(-a)3 = (+a3)X(—a) = —a3;

(—a)* = (—a3) ×(—a) = +a*;

(—a)* = (+a*)×(—a) = —a'.

And in general,

(-a)"=±a",

the plus sign in the second member being used when n is even, and the minus sign when n is odd. Hence,

1.-All powers of a positive quantity are positive.

2.-The odd powers of a negative quantity are negative, but the even powers are positive.

211. From the foregoing principles relating to the involution of a monomial, we derive the following

RULE. I. Raise the numeral coefficients to the required power.

II. Multiply the exponent of each letter by the exponent of the required power.

III. When the quantity involved is negative, give the odd powers the minus sign.