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How Do You Solve a Word Problem Using an AND Compound Inequality?

In order to get an 'A' in his Algebra class, Santiago needs to score at least 409 points on his final exam. He has done well throughout the semester, so he does not need to score more than 430 points to get the 'A'. Unfortunately though, Santiago is being deducted 200 points from his overall score for being late to class. If each question is worth 7 points, how many questions does Santiago need to answer correctly in order to get an 'A' in the class?

Summary

  1. We chose 'c' as our variable
  2. He needs at least 409 points but not more than 430
  3. '≤' means 'less than or equal to'
  4. A correct answer is 7 points, so '7c' represents the points for all his correct answers
  5. Subtract, or 'deduct', 200 points for being late, to get '7c-200'
  6. '≥' means 'greater than or equal to'
  7. We're dealing with a range so we have an 'AND' compound inequality
  8. Brackets symbolize a set, and '|' means 'such that'

Notes

    1. We chose 'c' as our variable because 'correct' begins with a 'c'
    1. Go back to the word problem and try to translate it to an equation so we can solve for 'c'
    2. Remember, 'c' is a variable representing the number of correct answers given
    1. So for every correct answer Santiago gives, he will earn 7 points!
    2. To find his overall score, we need to take 7 times the number of correct answers, 'c'!
    1. We know each question is worth 7 points
    2. Remember, 'c' is a variable representing the number of correct answers given
    3. 7c = total points from all the correct answers
    4. We have to subtract 200 points from this score because Santiago was late to class
    5. 'Subtract' means the same as 'deduct'
    6. So start with '7c-200'
    7. Our problem says he needs to score at least 409 points, but doesn't need more than 430
    8. Our equation now becomes:
    9. 409 ≤ 7c-200 ≤ 430
    10. '≤' means less than or equal to
    11. '≥' means greater than or equal to
    1. We have a compound inequality, which we can separate into two inequalities
    1. Our compound inequality was originally:
    2. 409 ≤ 7c-200 ≤ 430
    3. '≥' means greater than or equal to
    4. '≤' means less than or equal to
    1. Our compound inequality was originally:
    2. 409 ≤ 7c-200 ≤ 430
    3. This is an 'AND' compound inequality because his score will fall within a range
    4. '≤' means less than or equal to
    1. The Addition Property of Inequality says we can add 200 to both sides of the inequality
    2. Canceling the 200s will get us closer to having 'c' by itself
    1. The Division Property of Inequality says we can divide by 7 on both sides of the inequality
    2. 609 ÷ 7 = 87
    3. 630 ÷ 7 = 90
    1. Dividing by a negative number will flip the inequality symbol
    2. Since we divided by positive 7, the inequality symbol stays the same
    1. Dividing by a negative number would flip the inequality sign
    2. Since we divided by positive 7, the inequality symbol stays the same
    1. 'c' must be greater or equal to 87 but less than or equal to 90
    1. Our answer is that 'c' must be greater or equal to 87 but less than or equal to 90
    1. Brackets symbolize a set, and '|' means 'such that'
    1. Our two inequalities simplified to 'c≥87' and 'c≤90'
    2. Flipping the first inequality allows us to combine the two inequalities and write 'c' once
    3. '≤' means less than or equal to
    4. '≥' means greater than or equal to
    1. Our answer in set-builder notation is:
    2. { c | 87 ≤ c ≤ 90 }
    3. Remember, 'c' is a variable representing the number of correct answers given
    4. '≤' means less than or equal to