Find the maximum and minimum values of x in the

22. What values of x and y satisfy simultaneously the equation

and the inequality

2x-3y=2, 3x+4y>11.5?

By assigning to x any value greater than 2.5 and solving the equation for y, we have y greater than 1. (Show this.

of y.)

Let x = 4, and find value

Find values of x and y that will satisfy the following systems:

27. Show that the sum of any positive number n and its reciprocal is greater than 2, except when n=1.

If the literal numbers are unequal and positive, show that:

33. a3+1>a2+a, if a is a negative proper fraction.

a b

+<2, if a and b are of opposite qualities.

34.

b

a

36. a−b>(√ā — √b)2, if a>b.

37. Twice the number of pupils in a class diminished by 7 is greater than 29; and 3 times the number diminished by 5 is less than twice the number increased by 16. How many pupils in the class?

38. Five times a certain number diminished by 70 is less than 4 times the number diminished by 9; and 3 times the number increased by 2 exceeds twice the number by 61. What is the number?

39. A man does not have enough money to pay for a $20 watch, but if he borrows half as much as he has, he can pay for it and have more money left than he now lacks. How much money has he?

RATIO, PROPORTION, AND VARIATION

RATIO

394. The relation of one number to another of the same kind is Ratio, when it is expressed by the quotient of the first number divided by the second.

Thus, by the ratio of two algebraic numbers a and b is meant the quotient of a divided by b. Why is there no ratio between $3 and 6 hats? In algebra the letters used always represent abstract numbers.

395. The symbol of ratio is (:).

Thus, the ratio of a to b is written a b. But ratio may also be expressed in the fractional form, as

α

b

396. The first of the two numbers compared is called the Antecedent, the second the Consequent.

Thus, in the ratio a: b, a is the antecedent and b the consequent. It follows from the definition that the ratio of one number to another is found by dividing the antecedent by the consequent. Why is the ratio always an abstract number?

397. If the antecedent and consequent are interchanged, the resulting ratio is called the inverse of the given ratio.

Thus, the ratio ba is called the inverse of the ratio a: b.

398. If the ratio of two numbers is expressible as a rational number, the numbers are said to be Commensurable; otherwise they are said to be Incommensurable.

Thus, the diagonal and side of a square are incommensurable, their ratio being expressed by √2, an irrational number. Are the diameter and circumference of a circle commensurable?

399. PRINCIPLE. Multiplying or dividing both antecedent and consequent by the same number does not change the ratio.

a

For the ratio a: b is identical with and by Art. 176,

a

b

b

am

bm

= am : bm.

.. a: b =

Are all the properties of fractions true of ratios? Why?

2. If a<b, prove that the ratio of a: b is increased by adding any positive number, m, to both antecedent and consequent.

What is the effect of adding m to both antecedent and consequent when ab? When ab? If 2 is added to both numerator and denominator of , what is the effect on the value? If 2 is added to both numerator and denominator of 3, what is the effect on the value ? conclusion may be drawn from a comparison of these examples?

What

20. Divide 56 into two parts whose.ratio shall be 2:3.

21. What is the effect produced on the ratio 4:5 by multiplying the antecedent by 3? The consequent by 3? Both by 3?

22. If a is the antecedent and 2 the value of ratio, what is the consequent?

23. What number must be added to both antecedent and consequent of the ratio to make the ratio ?

24. What number must be subtracted from both antecedent and consequent of the ratio 3:5 to produce the ratio 5:9?

25. A certain ratio becomes when 4 is added to both antecedent and consequent, and when 2 is subtracted from each. Find the ratio.

26. If a >b, will the ratio a:b be increased or diminished by adding any positive number, m, to both antecedent and consequent ?

27. If a >b, will the ratio ab be increased or diminished by subtracting any positive number, m, not greater than b, from the antecedent and consequent ?