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First, It will be found that x is more than 4, and less than 5. Substituting these numbers for x, we have

x3

10x2
5x

64

160

20

244

68.921

168.10

20.5

257.521

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1

-15

63

49

400

Then, 400-244: 5-4 :: 260-244: the correction . 1.

This correction added to the less assumed number, gives 4.1 for an approximate value of x.

Secondly, By substituting 4.1 and 4.2 for x, we have

x3

10x2
5x

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-15.9135

64.89

50.069227

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First, Substituting 1 and 1.5 for x,

x3 -15x2. 63x

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-15.x2

63x

125

· 250

25

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Then, 271.488-257.521: 4.2-4.1 260-257.521: the correc tion .017.

Adding this correction to the less assumed number 4.1, we have 4.117 for a nearer value of x.

7. Find an approximate value of x in the Equation

x3-15x2+63x=50.

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Then, 64.125-49: 1.5—1:50-49: the correction .03.
Hence we have 1+.03=1.03, for an approximate value of x.
Secondly, Substituting 1.03 and 1.02 for x,

1.092727 .

x3.

3.375

-33.75

94.5

64.125.

1.061208

-15.606

64.26

49.715208.

Then, 50.069227-49.715208: 1.03-1.02: 50-49.715208: the "ection .00804.

rlence x 1.02+.008041.02804'.

8. Find an approximate value of x in the Equation

x3-17x2+54x=350.

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-3799.5425

807.30

349.119875

-17x2

54x

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10000.

-300

-750

8950

Then, 360-168:15-14:: 360-350: the correction .05. Subtracting this correction from the greater assumed member, we have 14.95 for an approximate value of x.

Secondly, Substituting 14.95 and 14.96 for x, we have

3341.362375 .

x3

-17x2

54.x

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9. Find an approximate value of x in the Equation

x4-3x2-75x=10000.

First, Substituting 10 and 11 for x, we have

x4

-3x2

752

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X4

-3.x2

-75x

807.84 351.284736.

Then, 351.284736-349.119875: 14.96-14.95: 350-349.119875: the correction .004065.

Adding this correction to the less assumed number 14.95, we find 14.954065 for a very near value of x.

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3375

-3825

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810

360.

-3804.6272

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3348.071936

Then, 13453-8950: 11-10 :: 10000-8950: the correction .23. Hence 10.23 is the first approximation to the value of x.

Secondly, Substituting 10.2 and 10.3 for x,

10824.3216

-312.12

-765

9747.2016

14641

-363

-825

13453.

11255.0881

-318.27

-772.5

10164.3181.

Then, 10164.3181-9747.2016: 10.3-10.2:: 10164.3181-10000: the correction .039393.

Hence 10.3-.039393=10.260607, is a nearer value of x.

10. Find an approximate value of x in the Equation

2x4-16x3+40x2-30x+1=0.

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-39

1

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--35.152

67.6

1

3

25

.1602

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-30x
1

31.622

56.622

Then, 5-(-3): 2-1::0-(-3): the correction .3.
Therefore, 1+3=1.3, is an approximate value of x.

Secondly, Substituting 1.3 and 1.2 for x,

5.7122

2x4

-16x3

40.x2

-30.x

1

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Substituting 10 and 11 for x,

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11. Find an approximate value of x in the Equation

2

x2-15) +x√x=90.

-15)2 +

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-.9008.

Then, .1602-(-9008): 1.3-1.2:: 0-(-9008): the correction .084. Hence 1.2+.084–1.284, is a nearer value of x.

1

5.

-27.648

57.6

-36

1

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4.1472

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84.64

36.482

121.122.

Then, 121.122-56.622:11-10 :: 121.122-90: the correction .48. Therefore 11.48 10.52, is an approximate value of x.

PROBLEMS IN 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.

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Let x represent the number; then we shall have

5x40, and 3x2+5>80.

Dividing the first Inequation by the coefficient of x, we find x<8. Transposing 5 in the second Inequation, and adding similar terms, 3x2>75; hence x2>25, and x>5.

The required number is therefore indeterminate within the limits 5 and 8; that is, it is either of the numbers 5, 6, cr 7, 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?

Let x represent the number; then we have

Xx

3

from which we shall find x>69, and <104.

X

-320, and 4<30;

4

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

Let x represent the number; then

x2-10<90; and √x+2>5.

These Inequations will give x<10, and >9; hence the number is 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.

Let x represent the number; then

X

X

x

2x+3x+4x100; and + + <30;

2 3

from which we shall find-x>114, and <27

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?

Let x represent the number; then

3x-6x+48; and 4x+6>27+3x;

from which x<27 and >21; hence the number of cattle was 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.

Let x represent the number; then

4x+x245; and x2-10x<75;

Completing the square in the first member of each of these Inequations, we have

x2+4x+4>49; and x2-10x+25<100.

Extracting the square root of each member,
x+2>7; and x-5<10.

from which x>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.

Let x represent the number of miles; then

5x+10100; and 8x-5x+100.

These Inequations will give x<18 and >15; therefore the number of miles was 16, or 17, or either 15, 16, or 17, plus any proper fraction.

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