Neft Teacher · Learn It

Unit 8 · Standard 6.AT.8

Equations and Inequalities Problem Solving

Lesson 8-7 · Learn It — how the math works

Why we're learning this

Detective Santos is closing a complex case. She has collected equations and inequalities from different parts of the investigation. One clue says: 'The total value of stolen items divided equally among 4 accomplices gave each person $85.' Another says: 'The lookout earned less than $50.' She must model both clues mathematically and check if the answers are reasonable. Can you help?

✱ Math Workbench

Learn It

Should I model this problem with an equation or an inequality?

To model means to write a real situation as math. I use an equation when there is one exact total, and an inequality when there is a limit or a range of allowed values.

Picture itSee what the math looks like before you read about it

Detective Chen is comparing two case notes. Note A says: 'The evidence locker holds exactly 54 files, split equally among 6 shelves.' Note B says: 'The surveillance van must stay parked for no more than 45 minutes.'
Where you see this in real lifeDetective Chen is comparing two case notes. Note A says: 'The evidence locker holds exactly 54 files, split equally among 6 shelves.' Note B says: 'The surveillance van must stay parked for no more than 45 minutes.'

Watch me solve itI do — read one step at a time

The problem looks like this
Tape diagram: 4 equal parts of $85 make the total t, and t ÷ 4 = 85 gives t = 340.t = 34085$85 each858585
  1. Clue: 'A total divided equally among 4 people gave each $85.'
  2. This has an exact total, so I use an . Let t be the total.
  3. Divided among 4 to get 85 means t ÷ 4 = 85, so t = 85 × 4 = 340.
  4. Check : $340 split 4 ways is $85 each. That makes sense!

Watch out — mistakes students make here

  • She gave donuts away, so subtract 7, don't add it.
  • 'Fewer than 5' means less than 5 — use '<', not '>'.
  • 'Fewer than' is an inequality, not equal — use '<'.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Clue: 'The lookout earned less than $50.' or ?
  2. 'Less than' is a limit, so it is an . Let e be the earnings.
  3. 'Less than 50' means e < 50. Is $30 a reasonable value?
    Check this step
    Yes.

On your ownYou do — solve this one by yourself

Before you start, remember:

  • In the table, decide or , write the , solve, and check it is reasonable.
  • Then work through the practice problems on your own.

Solve: Complete the table.

Word ProblemModelSolutionReasonable?
Five identical boxes weigh a total of 60 pounds. How much does each box weigh?
A shelf can hold at most 30 books. There are already 17 books. How many more can be added?
A baker split 84 cookies equally into bags of 12. How many bags did she use?

Stuck? Open one hint at a time.

Hint 1

For each row: find the numbers, decide the operation from the story's clue words, then ask if the answer makes real-world sense.

Hint 2

Equal groups (boxes, bags) point to a multiplication model; 'at most' points to an inequality with the 'or equal' bar. Solve with the inverse operation.

Hint 3

Row 1: five EQUAL boxes total 60 pounds — write a multiplication model with w, then divide both sides by 5. Is that weight per box sensible?

Check my answer
✅ Answers:
  • Five identical boxes weigh a total of 60 pounds. How much does each box weigh? → Model: 5w = 60
  • Five identical boxes weigh a total of 60 pounds. How much does each box weigh? → Solution: w = 12 pounds
  • Five identical boxes weigh a total of 60 pounds. How much does each box weigh? → Reasonable?: Yes — 12 pounds per box is reasonable.
  • A shelf can hold at most 30 books. There are already 17 books. How many more can be added? → Model: 17 + b ≤ 30
  • A shelf can hold at most 30 books. There are already 17 books. How many more can be added? → Solution: b ≤ 13 books
  • A shelf can hold at most 30 books. There are already 17 books. How many more can be added? → Reasonable?: Yes — up to 13 more books fits.
  • A baker split 84 cookies equally into bags of 12. How many bags did she use? → Model: 12b = 84
  • A baker split 84 cookies equally into bags of 12. How many bags did she use? → Solution: b = 7 bags
  • A baker split 84 cookies equally into bags of 12. How many bags did she use? → Reasonable?: Yes — 7 bags of 12 = 84 cookies.
Each row goes story → model → solution → reasonableness check. Five equal boxes totaling 60 pounds is 5w = 60, so w = 12 pounds. 'At most 30 books' with 17 already on the shelf is 17 + b ≤ 30, so b ≤ 13 books. And 84 cookies split into bags of 12 is 12b = 84, so b = 7 bags. Each answer is checked by putting it back into the story: 7 bags of 12 really is 84 cookies.

Turn & TalkSay it out loud with a partner

Before solving a word problem, how do you decide whether to it with an or an ?

Try starting with:

  • I would model this with a(n) ___ because the problem says ___.Modelaría esto con un(a) ___ porque el problema dice ___.
  • Words like ___ tell me to use ___.Palabras como ___ me dicen que use ___.
Use these wordsmodelequationinequalityreasonableness