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Dividing both members by -5 (Art. 223),

y<5.

Find the limits of x in the following:

3. (6x+1)- 105 < (4x-3) (9x+4).

4. (2x+3)(3x-1)> (2x+7) (3x-2)+1.

5. (x+1)(x+2) (x − 3) > (x − 1) (x − 4) (x+5).

6. 3 ax+14 ab>6a2+7bx, if 3a-7b is a negative number.

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Find the limits of x and Y in the following:

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2x-921-4x, and 3x - 11 > 5x — 41.

11. A certain positive integer, plus 23, is less than 6 times the number, minus 12; and 9 times the number, minus 54, is less than twice the number, plus 9. What is the number?

12. A teacher being asked the number of his pupils, replied that 29 was less than twice their number, diminished by 7; and that 5 times their number, diminished by 5, was less than twice their number, increased by 55. Required the number of his pupils.

13. A shepherd has a number of sheep such that twice. the number, diminished by 45, exceeds 79, diminished by twice the number; and 5 times the number, increased by 1, is less than 3 times the number, increased by 69. How many sheep has he?

or,

14. Prove that if a and b are positive numbers,

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Since the square of any number is positive (Art. 109),

That is,

(a - b)>0.

a2-2ab+b2 > 0,

a2 + b2 > 2 ab.

Dividing each term of the inequality by ab (Art. 222),

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15. Prove that, for any value of x, 4x2+9 is not less than 12x.

5x

16. Prove that, for any value of x, x2+ is not less than 5x

25
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17. Prove that, for any values of a and b, (2a+b)(2 a−b) is not less than 2b(6a — 5b).

18. Prove that, for any values of a, b, c, and d, (ac — bd)2 is not less than (a - b) (c-d2).

19. Prove that, if a and b are positive numbers,

a3 +b3> a2b+ ab2.

20. Prove that, if a2 + b2 = 1 and c2 + d2 = 1, then

ab + cd < 1.

XV. INVOLUTION.

228. Involution is the process of raising a given expression to any power whose exponent is a positive integer. This may be effected by taking the product of as many expressions, each equal to the given expression, as there are units in the exponent of the required power (Art. 8).

229. We have already given (Art. 109) a rule foi · raising a rational and integral monomial to any power whose exponent is a positive integer.

A fraction may be raised to any power whose exponent is a positive integer by raising both numerator and denominator to the required power, and dividing the first result by the second.

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230. Square of a Polynomial.

We gave in Art. 108 rules for the square of a binomial.

Thus,

(α1 + α)2 = α12 + a22+27α.

We also find by multiplication:

(1)

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The results (1) and (2) are in accordance with the following law:

The square of a polynomial is equal to the sum of the squares of its terms, plus twice the product of each term by each of the following terms.

We will now prove by Induction (Note, p. 46) that this law holds for the square of any polynomial.

Assume that the law holds for the square of a polynomial of m terms, where m is any positive integer; that is,

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This result is in accordance with the above law.

Hence if the law holds for the square of a polynomial of m terms, where m is any positive integer, it also holds for the square of a polynomial of m+1 terms.

But we know that the law holds for the square of a polynomial of three terms, and therefore it holds for the square of a polynomial of four terms; and since it holds for the square of a polynomial of four terms, it also holds for the square of a polynomial of five terms; and so on.

Hence the law holds for the square of any polynomial.

Example. Expand (2x2 - 3x — 5)2.

In accordance with the above law, we have

(2x2-3x-5)2

=(2x2)2+(-3x)2 + ( − 5)2

+2(2x)(-3x)+2(2x2) (→ 5) + 2(−3x) ( − 5) =4x+9x2+25-12-20x2+30x

=4x-12x3-11 x2+30 x + 25.

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That is, the cube of the sum of two quantities is equal to the cube of the first, plus three times the square of the first times the second, plus three times the first times the square of the second, plus the cube of the second.

The cube of the difference (Art. 108, Note) of two quantities is equal to the cube of the first, minus three times the square of the first times the second, plus three times the first times the square of the second, minus the cube of the second.

1. Find the cube of a +26.

(a + 2b)3 = a3 +3a2 (2b)+3a (2b)2 +(2b)3
= a3+6a2b+ 12 ab2 + 8 b3.

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