EXAMPLE III.

To find a number such, that if 82 be increased by 3 times that number, of the sum will be equal to 133.

If x represent the number, the equation will be

Hence no number can be found that will fulfil the conditions of the problem; that is, the problem is impossible. The result shows that of 82 itself is equal to 133.

EXAMPLE IV.

of it and of it, dimi

To find a number such, that the sum of nished by 2, shall be equal to of it increased by 3.

If a represent the number, the equation will be

From this equation we shall find

3x+8x-24=11x+36; or 11x-11x=0x=60;
and x=60, infinity; (50).

The result shows that it would require a number infinitely great, to fulfil the conditions of the problem. The problem is therefore impossible.

EXAMPLE V.

To divide the number 24 into two such parts, that their product shall be 150.

If x represent one of the two parts, 24-x will represent the other and the equation will be

24x-x2-150

or x2-24x=-150, (117);

which gives x12±√/144-150,

=12±√6.

In this value of x, the part V-6 is imaginary, that is, it is an impossible quantity, (246); hence the problem is impossible.

That the preceding problem is impossible, will also appear from the following general proposition; viz:

(273.) The square of one half of any quantity is greater than the product of any two unequal parts of the quantity.

Let s represent any number, and d the difference between any two parts of the number; then the respective parts are

This product will vary directly as its numerator s2-d2 (147); and will therefore have its greatest value when d=0. In that case the product becomes s2, which is the square of s, half the given number.

It

may also be remarked here, that the sum of the two equal factors of any quantity, is less than the sum of any two unequal factors into which the quantity can be resolved.

For, s representing the sum, and d the difference, of the two factors of a quantity, those factors are

s+d s-d

2

and (168), and their product is

2

s2-d2
4

This product will retain a constant value, if s2 and d2 be equally drminished, for then the value of s2-d2 will remain constant. In this diminution, s2 will have its least value when d2 is made 0; so that s, the sum of the two factors, will have its least value when a=0, and the two factors will then be equal to each other.

Signification of the Different Forms under which the Value of the Unknown Quantity may be found in an Equation.

(274.) 1. Positive values of the unknown or required quantities, fulfil the conditions of problems in the sense in which they are proposed.

2. A value of the unknown quantity of the form %, shows that the problem from which the equation was derived is indeterminate.

3. A negative value of the unknown quantity, in an equation of the first degree, indicates an impossibility in the problem, produced by taking this quantity additively, instead of subtractively, or vice versa.

4. When the value of the unknown quantity in an equation is zero, infinite, or imaginary, the problem from which the equation was derived is impossible.

10*

INEQUATIONS.

(275.) An INEQUATION is an expression of the inequality between two quantities by means of the sign >, greater than, or <, less than, placed between them.

Thus a>x+y denotes that a is greater than x+y;

x-y<b denotes that x-y is less than b.

It will be observed that the sign opens towards the greater quantity.

(276.) Inequations are employed in the solution of Problems, whose conditions involve inequalities, and render the required quantities indeterminate within certain necessary limits.

They are also sometimes employed to determine the sense in which an inequality between given quantities subsists.

(277) Two Inequalities are said to subsist in the same sense, when the greater quantity stands on the right of the sign in both, or on the left in both; otherwise, the Inequalities subsist in a contrary sense.

In a>b and y>x the Inequalities subsist in the same sense, in ab and x<y the Inequalities subsist in a contrary sense

Inequalities between Negative Quantities.

(278.) Any disconnected negative quantity, as -a, may be regarded as 0-a; and of two such quantities, that one is therefore the less, algebraically, which, if both were positive, would be the greater.

Thus -5 is, algebraically, less than -3; that is, 0-5 is less than 0-3. The consistency of this will appear from considering, that the greater the quantity subtracted, the less will be the remainder.

A negative quantity being thus regarded as the remainder when the corresponding positive quantity is subtracted from 0, is sometimes said to be less than 0, and therefore less than any positive quantity. This must be understood, however, as merely implying the contrariety of a negative to a positive quantity, in its effect upon a calculation; for a quantity, considered in itself, can never be less than nothing.

Transformation of Inequations.

(279.) An Inequation may be transformed by clearing it of fractions, by the transposition and addition of terms, &c., in the same manner as an Equation; and the result will generally preserve the Inequality in the same sense.

But,

I. If both sides of an Inequation be multiplied or divided by a nega tive quantity, the resulting Inequality will subsist in a contrary sense.

Thus by multiplying both sides of the Inequation 5>3 by-2, we have the contrary Inequality-10<-6. (278.)

II. If the signs of all the terms of an Inequation be changed, to and to +, the Inequality will be changed to the contrary sense.

This follows from the preceding proposition, since the changing of all the signs, + and is equivalent to multiplying both sides of the

Inequation by 1.

III. Like powers or roots of two sides of an Inequation will sometimes form an Inequality subsisting in à contrary sense.

Thus by squaring both sides of -3>-5 we have 9<25, in which the Inequality subsists in a contrary sense.

And by extracting the square root of both sides of 25>16, we have 5>4, or - 5-4, in the latter of which the Inequality is changed to the contrary sense.

IV. If the corresponding sides of two Inequations, subsisting in the same sense, be subtracted the one from the other, the result will sometimes form an Inequality in the contrary sense.

Thus by subtracting 10>5 from 12>9, we have 2<4, in which the Inequality subsists in a contrary sense.

An Example of the use of Inequations may be given in showing that the sum of the squares of any two unequal quantities is greater than twice the product of the quantities.

Since every even power of a quantity is positive, we have

(a-b)2 or a2-2ab+b2>0.

Adding 2ab to each side of the Inequation, we find that
a2+b2>2ab.

Exercises on Inequations.

1. Find a number such, that when multiplied by 5, the product shall be less than 40, and when 5 is added to 3 times the square of the number, the sum shall be greater than 80.

5x40, and 3x2+5>80.

Proceeding in the same manner as with Equations, we shall find a to be <8, and >5.

The required number is therefore indeterminate within the limits 8 and 5; that is, it is 7, or 6, or either of these numbers plus any proper fraction.

2. What number is that whose third part diminished by 3 is greater than 20, and whose fourth part increased by 4 is less than 30?

Ans. Any number between 69 and 104.

3. Find a number whose square diminished by 10 is less than 90, and whose square root increased by 2 is greater than 5.

Ans. 9 plus any proper fraction.

4. Find a number such, that if it be multiplied by 2, 3, and 4, successively, the sum of the products shall be greater than 100, and if it be divided by the same numbers, the sum of the quotients shall be less than 30.

Ans. Any number between 113 and 271

5. A farmer sold a number of cattle. If 6 be subtracted from 3 times the number, the remainder will be less than the number increased by 48; and if 6 be added to 4 times the number, the sum will be greater than 27 increased by 3 times the number; what was the number of cattle?

Ans. 22, 23, 24, 25, or 26.

6. Find a number whose square added to 4 times the number itself shall be more than 45, but whose square diminished by 10 times the number shall be less than 75.

Ans. Any number between 5 and 15.

7. Five times the number of miles between two places increased by 10, is less than 100; and 8 times the number diminished by 5, is greater than the number increased by 100. Required the number of miles between the two places.

Ans. Any number between 15 and 18.