Elementary Algebra and Geometry for Schools

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Brackets, braces, and parentheses.

Brackets [ ] and braces { } are used, as are parentheses, to indicate that an expression is to be considered as a single quantity.

We know that 3x -(a + b) means that the sum of a and b is to be subtracted from 3x. How may we show that this result, or difference, is to be subtracted from another quantity, for example 5a? To consider 3x -(a + b) as a single quantity, we will enclose it in brackets: [-3x -(a + b)]. Braces would serve equally well. To indicate that [-3x -(a + b)] is subtracted from 5a, we write: 5a-[3x-(a+b)]. The use of brackets or braces instead of another set of parentheses prevents confusion which might result from having two sets of parentheses, one set within the other.

ILLUSTRATIVE EXAMPLES: Removing parentheses within brackets

1. Simplify the expression 2x -[5x -(x -2a)].

Solution. Given 2x -[5x -(x -2a)]

Since the parentheses are contained within the brackets, we first remove the parentheses 2x -[5x -x + 2a]

Collecting like terms within the brackets

2x [-4x +2a]

Removing the brackets

2x-4x-2a

Collecting like terms

-2x -2a

2. Simplify the expression                x2-[y2 -{3x2-(2x2+3y2) +5x2}-4y2]

Solution. Given                               x2-[y2 -{3x2-(2x2+3y2) +5x2}-4y2]

Removing parentheses                    x2-[y2 -{3x2-2x2-3y2 +5x2}-4y2]

Collecting like terms within braces     x2-[y2 -{6x2-3y2}-4y2]

Removing braces                            x2-[y2 -6x2+3y2-4y2]

Collecting like terms within brackets  x2-[-6x2]

Removing brackets                          x2+ 6x2

 In simplifying algebraic expressions where symbols of grouping appear within other symbols of grouping, remove the symbols of grouping one at a time beginning with the innermost set.

EXERCISES: Removing parentheses, brackets, braces

Simplify each of the following: • 1. x-a-b
• 2. 2a+2b
• 3. a -3b
• 4. x -y
• 5. y
• 6. a + b
• 7. 2x2
• 8. 4y+1
• 9. 4x-5
• 10. a + 3b
• 11. x2 -6x
• 12. x -y
• 13. a -5b
• 14. x-a
• 15. 3b -3a
• 16. -2x -3y
• 17. -4a-4b
• 18. 7x+y-2
• 19. 4a -a2+b2
• 20. 7y2 -3y

EXERCISES: Solving equations containing parentheses, brackets, and braces

Solve each of the following for the unknown letter: 1. 2; 2. 5; 3. 1. 4. -2; 5. -1; 6. 1; 7. 4; 8. 3; 9. 1; 10. 5.

Representing two numbers in terms of a single unknown letter when the sum of the numbers is given.

The sum of two numbers is 18. If one of the numbers is 7, what is the other number? You undoubtedly found that the other number was 11, but think back for a moment to see exactly how you did obtain this answer. What you did, of course, was to subtract the given number 7 from the given sum 18. In other words, you can represent, in this case, the other number in terms of the known numbers as 18 -7.

Suppose that the sum of two numbers is 40. If one of the numbers is x, how can you represent the other number? Just as in the problem above, you can represent the other number by subtracting the number x from the sum of the two numbers, 40. In other words, you can represent the other number as 40-x

ILLUSTRATIVE EXAMPLES: Representing a number in terms of a given number x

• 1. The sum of two numbers is 20. If x represents one of the numbers, how would you represent the other number in terms of x?
• Solution. The other number can be obtained by subtracting the number x from the given sum, 20; hence, the other number can be represented as 20-x.
• 2. The sum of two numbers is 35. If x represents the larger number, how would you represent the smaller number?
• Solution. The smaller number can be obtained by subtracting the number x from the given sum, 35; hence the smaller number can be represented as 35-x.
• 3. The sum of two numbers is s. If x represents one of the numbers, how would you represent the other number in terms of s and x?
• Solution. The other number can be obtained by subtracting the number x from the sum s; hence the other number can be represented as s-x.

ORAL EXERCISES

• 1. The sum of two numbers is 100. If the larger number is represented by x, how would you represent the smaller number?
• 2. The sum of two numbers is 86. If one of the numbers is represented by y, how would you represent the other number?
• 3. The sum of two numbers is 8. If x represents one of the numbers, how would you represent the other number?
• 4. The sum of two numbers is 93. If a represents one of the numbers, how would you represent the other number?
• 5. The sum of two numbers is 104. If n represents one of the numbers, how would you represent the other number?
• 6. Two boys together have 167 marbles. If the number of marbles that one boy has is represented by x, how would you represent the number of marbles the other boy has?
• 7. A man invests \$10,000 in stocks and bonds. If the number of dollars that he invests in stocks is represented by x, how would you represent the number of dollars that he invests in bonds?
• 8. A man buys 100 stamps, some of them 3-cent stamps and the rest 5-cent stamps. If the number of 3-cent stamps that he buys is represented by x, how would you represent the number of 5-cent stamps that he buys?
• 9. The sum of the ages of a girl and her father is 51 years. If the number of years in the girl's age is represented by x, how would you represent the number of years in the father's age?
• 10. A boy has 73 coins consisting of nickels and dimes. If the number of nickels that the boy has is represented by x, how would you represent the number of dimes that he has?

• 1. 100-x
• 2. 86-y
• 3. 8-x
• 4. 93-a
• 5. 104-n
• 6. 167-x
• 7. 10,000 -x
• 8. 100-x
• 9. 51-x
• 10. 73-x.

COMPLETION EXERCISES

On another sheet of paper write the numbers of the incomplete statements in a column. Opposite the number of each incomplete statement write the word for each blank which will make the incomplete statement complete and true.

• 1. The sum of two numbers is 83. The larger number can be represented by -- and the smaller number by --.
• 2. The sum of two numbers is 74. The smaller number can be represented by -- and the larger number by -- .
• 3. The sum of the ages of a boy and his sister is 33 years. The number of years in the boy's age can be represented by -- and the number of years in the sister's age can be represented by -- .
• 4. A man invests \$3000 in stocks and bonds. The number of dollars that he invests in bonds can be represented by -- and the number of dollars that he invests in stocks can be represented by --.
• 5. A man has 57 coins composed of quarters and dimes. The number of quarters that he has can be represented by -- and the number of dimes that he has can be represented by --.
• 6. The sum of the ages of Mary and Nancy is 36 years. The number of years in Mary's age can be represented by -- and the number of years in Nancy's age can be represented by --.
• 7. The cost of a table and a chair is \$240. The cost of the table (in dollars) can be represented by -- and the cost, in dollars, of the chair can be represented by --.
• 8. The distance covered by a man on a trip was 450 miles. If he drove part of the way and traveled the rest of the way by train, then the number of miles he drove can be represented by -- and the number of miles he traveled by train can be represented by --.

• 1. x, 83 -x
• 2. x, 74 -x
• 3. x, 33 -x
• 4. x, 3000 -x
• 5. x, 57 -x.
• 6. x, 36 -x
• 7. x, 240 -x
• 8. x, 450 -x.

ILLUSTRATIVE EXAMPLE: Solving word problems

The sum of two numbers is 31. Three times the larger increased by twice the smaller is equal to 79. Find the two numbers.

Solution. Since we know the sum of the two numbers, we can represent each of the numbers in terms of a single unknown x: Let the larger number be represented by x; then the smaller number can be represented by 31 -x.

We translate the second sentence into an equation: Answering the question, the larger number, x, is 17; the smaller number. 31 -x, is 31-17, or14.

Check. Is the sum of the two numbers, 17 and 14, equal to 31? Is three times the larger, 51 (three times 17), increased by twice the smaller, 28 (twice 14), equal to 79? Since the answer to both of these questions is yes, the numbers 17 and 14 satisfy the statement of the problem.

EXERCISES: Word problems

• 1. The sum of two numbers is 43. Twice the larger increased by three times the smaller is equal to 104. Find the two numbers.
• 2. The sum of two numbers is 13. The sum of six times the smaller and three times the larger is 54. Find the two numbers.
• 3. The sum of two numbers is 32. If twice the larger is subtracted from five times the smaller, the result is 34. Find the two numbers.
• 4. The sum of two numbers is 27. Three times the larger, diminished by eight times the smaller, is equal to 4. Find the two numbers.
• 5. The sum of two numbers is 60. Five times the smaller diminished by twice the larger is equal to 41. Find the two numbers.
• 6. The sum of two numbers is 28. Three times the larger exceeds six times the smaller by 3. Find the two numbers.

• 1. 25, 18
• 2. 5, 8
• 3. 14, 18
• 4. 20, 7
• 5. 23, 37
• 6. 19, 9

Consecutive integers.

An integer is a "whole" number, such as 1, 2, 3 15, 84, 9, 17, etc. Consecutive integers are integers which follow each other in order; that is, 4, 5, 0, and 7 are consecutive integers; 11 and 12 are consecutive integers; 234, 235, and 236 are consecutive integers, etc.

We notice that to find each successive integer, we add one to the integer that precedes it. To find the integer that follows 9, we add 1 to 9 and obtain 10. In a similar manner, to find the integer which follows x (assuming x is an integer), we add 1 to x and obtain x +1. Thus if x represents an integer then x + 1 represents the next consecutive integer. How would we represent the next consecutive integer following x+1? We would add 1 to x+1. We write x+1+1, or more simply x+2.

Consecutive even integers are even integers which follow in order; that is, 2, 4, and 6 are consecutive even integers; 34 and 36 are consecutive even integers; 10, 12, 14, and 16 are consecutive even integers, etc.

We notice that to find each successive even integer, we add 2 to the even integer that precedes it. To find the even integer that follows 46, we add 2 to 46 and obtain 48. In a similar manner, to find the even integer which follows x (assuming x is an even integer), we add 2 to x and obtain x+2. Thus if x represents an even integer then x+2 represents the next consecutive even integer. How would we represent the next consecutive even integer following x+2? We would add 2 to x+2. We write x+2+2, or more simply x+4.

Consecutive odd integers are odd integers which follow in order; that is, 3,5, and 7 are consecutive odd integers; 21,23,25, and 27 are consecutive odd integers; 83 and 85 are consecutive odd integers; etc.

We notice that to find each successive odd integer, we add 2 to the odd integer that precedes it. To find the odd integer that follows 57, we add 2 to 57 and obtain 59. In a similar manner, to find the odd integer which follows x (assuming x is an odd integer), we add 2 to x and obtain x+2.

Thus if x represents an odd integer then x+2 represents the next consecutive odd integer. How would we represent the next consecutive odd integer following x+2? We would add 2 to x +2. We write x+2+2, or more simply x+4.

EXERCISES: Consecutive integers

• 1. Write four consecutive integers starting with 15.
• 2. Write three consecutive even integers starting with 20.
• 3. Write five consecutive odd integers starting with 5.
• 4. If x represents an integer, how would you represent the next two integers that follow x?
• 5. If x represents an integer, how would you represent the integer that precedes x?
• 6. If x represents an even integer, how would you represent the next consecutive even integer following x?
• 7. If x represents an odd integer, how would you represent the next consecutive odd integer following x?
• 8. How would you represent three consecutive integers if the smallest is represented by the letter x?
• 9. If x+6 is an even integer, how would you represent the next consecutive even integer following x+6?
• 10. Three consecutive even integers are represented as x, x+2, and x+4. How would you represent the sum of these three even integers?
• 11. If x represents an even integer, how would you represent the first odd integer following x? How would you represent the first even integer following x?
• 12. How would you represent three consecutive odd integers if the smallest is represented by the letter x?
• 13. How would you represent five consecutive even integers if the smallest is represented by the letter x?
• 14. If x-8 represents an odd integer, how would you represent the next consecutive odd integer?
• 15. How would you represent three consecutive integers if the smallest is represented by the letter x? How would you represent the sum of these three consecutive integers?

• 1. 15, 16, 17, 18
• 2. 20, 22, 24
• 3. 5, 7, 9, 11, 13
• 4. x + 1, x + 2.
• 5. x -1
• 6. x + 2.
• 7. x + 2
• 8. x, x + 1, x + 2
• 9. x + 8
• 10. 3x + 6
• 11. x + 1, x + 2
• 12. x, x + 2, x + 4
• 13. x, x + 2, x + 4, x + 6, x + 8
• 14. x -6
• 15. x, x + 1, x + 2; 3x + 3.

ILLUSTRATIVE EXAMPLES: Consecutive integers

1. Find the three consecutive integers whose sum is 72.

Solution. The three consecutive integers can be represented as: x, x +1, and x +2. Since their sum is 72, we add the three integers and set this expression equal to 72:

x+x+1+x+2 = 72

Collecting                                          3x +3 = 72

Subtracting 3 from both members             3x = 69

Dividing by                                              3 x = 23

Answering the question: the three integers are 23, 24, and 25.

2. Find three consecutive even integers such that five times the first, increased by twice the third, is equal to 106.

Solution. The three consecutive even integers can be represented as:

first even integer x

second even integer x+2

third even integer x+4 EXERCISES: Consecutive integer problems

• 1. Find three consecutive even integers whose sum is 96.
• 2. Find four consecutive even integers whose sum is 60.
• 3. Find three consecutive odd integers whose sum is 189.
• 4. Find three consecutive integers such that the sum of the first and third is 98.
• 5. Find two consecutive odd integers whose sum is 164.
• 6. Find two consecutive even integers such that five times the first added to three times the second gives a sum of 230.
• 7. Find three consecutive integers such that three times the last, subtracted from four times the first equals 12.
• 8. Find three consecutive even integers such that three times the first exceeds the sum of the second and third by 46.
• 9. Find three consecutive odd integers such that three times the last exceeds twice the first by 25.
• 10. Find four consecutive integers such that twice the sum of the last two integers exceeds three times the first integer by 91.

• 1. 30, 32, 34
• 2. 12, 14, 16, 18
• 3. 61, 63, 65
• 4. 48, 49, 50
• 5. 81, 83
• 6. 28, 30
• 7. 18, 19, 20
• 8. 52, 54, 56
• 9. 13, 15, 17
• 10. 81, 82, 83, 84

REVIEW EXERCISES

1. Find the sum of the following: 3x - 2, 5x +1, and 7 -4x.

2. Subtract 3x -5 from 8 +2x.

3. From the sum of 2x -5y and 6x +y subtract 3x -4y.

4. By how much does 2a +b exceed a b?

5. Find the sum of each of the following: 6. In each of the following subtract the lower polynomial from the one above it: 7. From the sum of 4a -5b +c and 6c -a -2b subtract the sum of a+b+c and 8a-7b.

8. Express algebraically: "a diminished by the sum of b and c."

9. Express algebraically: "z increased by twice the sum of x and y."

10. Simplify each of the following:

a) a-(b-a)+2(a -b)

b) 3x-[2y-(x+y)]+2y

11. The sum of two numbers is 47, and three times the smaller number is equal to the larger number increased by 1. Find the two numbers.

12. Find three consecutive odd integers such that twice the first added to three times the second gives a sum that is 1 less than four times the third.

• 1.  4x+6
• 2.  -x+13
• 3.  5x
• 4.  a + 2b
• 5.  6x2 -X + 1, 4a3 -a2 -6a
• 6.  x2 -13xy, 14y3 -14y2 -18y + 10
• 7.  -6a -b + 6c
• 8.  a -(b + c)
• 9.  z + 2(x + y)
• 10.   (a) 4a -3b; (b) 4x + y
• 11.   12; 35
• 12.   9; 11; 13

CUMULATIVE REVIEW EXERCISES 8. Express algebraically:

• a) 6 increased by 2a
• b) x decreased by y
• c) nine times the sum of 2x and y

9. The number of feet in the length of a rectangle is represented by 4x -7, and the number of feet in the width of the rectangle is 3x feet less than the length. Express the perimeter of the rectangle in terms of x.

10. Twice a number, increased by 12, is equal to five times the same number, decreased by 6. Find the number.

• 1.  (a) 2x3 -6x2y; (b) 17; (c) -6a5
• 2.  (a) -2x2; (b) -8x6; (c) 3y
• 3.  (a) 3; (b)- 6; (c) 3
• 4.  (a) 4a2; (b) x-9
• 5.  x3 -12x -12
• 6.  (a) 3; (b) 5
• 7.  (a) 0; (b) -1.
• 8.  (a) 6+2a; (b) x-y; (c) 9(2x + y)
• 9.  10x -28
• 10.  6