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Consecutive integers have the property that if

n = the smallest integer, then n + 1 = the next integer, and n + 2 = the next integer, and so on.

Consecutive odd or even integers have the property that if

n = the smallest integer, then n + 2 = the next odd or even integer (since odd or even numbers differ by 2), and n + 4 = the next odd or even integer, and so on.

The sum of three consecutive odd integers is equal to one less than twice the first odd integer. Find the three integers.

  1. Let n = the first odd integer. Then, n + 2 = the second odd integer, and n + 4 = the third odd integer.
  2. Translate the words to mathematical symbols and construct an equation. Read phrases.

    The sum of: add some numbers three consecutive odd integers: n , n + 2, n + 4 is equal to: = one less than: subtract 1 from twice the first odd integer: 2 n } n + ( n + 2 ) + ( n + 4 ) = 2 n 1 size 12{ left none matrix { "The sum of:" {} # "add some numbers" {} ##"three consecutive odd integers:" {} # n,n+2,n+4 {} ## "is equal to:" {} # ={} {} ##"one less than:" {} # "subtract 1 from" {} ## "twice the first odd integer:" {} # 2n{}} right rbrace n+ \( n+2 \) + \( n+4 \) =2n - 1} {}

  3. n + n + 2 + n + 4 = 2 n 1 size 12{n+n+2+n+4=2n - 1} {} 3 n + 6 = 2 n 1 size 12{3n+6=2n - 1} {} Subtract 2 n size 12{2n} {} from  both  sides. 3 n + 6 2 n = 2 n 1 2 n size 12{3n+6 - 2n=2n - 1 - 2n} {} n + 6 = 1 size 12{n+6= - 1} {} Subtract 6 from  both  sides. n + 6 6 = 1 6 size 12{n+6 - 6= - 1 - 6} {} n = 7 size 12{n= - 7} {} The first integer is -7. n + 2 = 7 + 2 = 5 size 12{n+2= - 7+2= - 5} {} The second integer is -5. n + 4 = 7 + 4 = 3 size 12{n+4= - 7+4= - 3} {} The third integer is -3.

  4. Check this result.
    The sum of the three integers is

    7 + ( 5 ) + ( 3 ) = 12 + ( 3 ) = 15

    One less than twice the first integer is 2 ( 7 ) 1 = 14 1 = 15 size 12{2 \( - 7 \) - 1= - "14" - 1= - "15"} {} . Since these two results are equal, the solution checks.

  5. The three odd integers are -7, -5, -3.
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Practice set a

When three times a number is decreased by 5, the result is -23. Find the number.

  1. Let x =
  2. Check:
  3. The number is .

-6

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When five times a number is increased by 7, the result is five less than seven times the number. Find the number.

  1. Let n =
  2. Check:
  3. The number is .

6

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Two consecutive numbers add to 35. Find the numbers.

  1. Check:
  2. The numbers are and .

17 and 18

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The sum of three consecutive even integers is six more than four times the middle integer. Find the integers.

  1. Let x = smallest integer.
    = next integer.
    = largest integer.
  2. Check:
  3. The integers are , , and .

-8, -6, and -4

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Geometry problems

Sample set b

The perimeter (length around) of a rectangle is 20 meters. If the length is 4 meters longer than the width, find the length and width of the rectangle.

  1. Let x = size 12{x={}} {} the width of the rectangle. Then,
    x + 4 = size 12{x+4={}} {} the length of the rectangle.
  2. We can draw a picture.

    A rectangle with base length the quantity x + 4 and height x.

    The length around the rectangle is
    x width + x + 4 length + x width + x + 4 length = 20

  3. x + x + 4 + x + x + 4 = 20 size 12{x+x+4+x+x+4="20"} {} 4 x + 8 = 20 size 12{4x+8="20"} {} Subtract 8 from  both  sides. 4 x = 12 size 12{4x="12"} {} Divide  both  sides by 4. x = 3 size 12{x=3} {} Then, x + 4 = 3 + 4 = 7 size 12{x+4=3+4=7} {}

  4. Check:

    A rectangle with base length 7 and height 3.

  5. The length of the rectangle is 7 meters.
    The width of the rectangle is 3 meters.
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Practice set b

The perimeter of a triangle is 16 inches. The second leg is 2 inches longer than the first leg, and the third leg is 5 inches longer than the first leg. Find the length of each leg.

  1. Let x = size 12{x={}} {} length of the first leg.
    = length of the second leg.
    = length of the third leg.
  2. We can draw a picture.
  3. Check:
  4. The lengths of the legs are , , and .

3 inches, 5 inches, and 8 inches

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Exercises

For the following 17 problems, find each solution using the five-step method.

What number decreased by nine is fifteen?

  1. Let n = size 12{n={}} {} the number.
  2. Check:
  3. The number is .

24

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What number increased by twelve is twenty?

  1. Let n = size 12{n={}} {} the number.
  2. Check:
  3. The number is .
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If five more than three times a number is thirty-two, what is the number?

  1. Let x = size 12{x={}} {} the number.
  2. Check:
  3. The number is .

9

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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