Lesson 8.7Equations and Inequalities Problem Solving
Start hereWords, worked example, and sentence starters
Learning target I can model and solve real-world problems using equations and inequalities.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| ModelSpanish: Modelar | To show a real-life situation with an equation or inequality. | 'Twice a number is 18' → 2n = 18 — the equation models the words |
| EquationSpanish: Ecuación | A math sentence with an equal sign showing both sides are the same. | x + 8 = 15 — exactly one answer: x = 7 |
| InequalitySpanish: Desigualdad | A math sentence that compares two sides with <, >, ≤, or ≥. | x + 5 > 12 — many answers: x = 8, 9, 10, ... |
| ReasonablenessSpanish: Razonabilidad | Checking if your answer makes sense. | If the problem asks for a number of people and you get x = −3, that is NOT reasonable |
| VariableSpanish: Variable | A letter that stands for a number that is unknown or can change. | “5 less than a number is 12” becomes x − 5 = 12, where x is the variable. |
| ConstraintSpanish: Restricción | A limit in a problem that tells what values are allowed. | “You can spend at most $20” is a constraint: cost ≤ 20. |
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Clue: 'A total divided equally among 4 people gave each $85.'
- This has an exact total, so I use an equation. Let t be the total.
- Divided among 4 to get 85 means t ÷ 4 = 85, so t = 85 × 4 = 340.
- Check reasonableness: $340 split 4 ways is $85 each. That makes sense!
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Clue: 'The lookout earned less than $50.' Equation or inequality?
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- I use an equation when the problem says , but I use an inequality when the problem says because .
Word bank ModelEquationInequalityReasonablenessVariableConstraintequationinequalityequalat leastat mostmore than
Watch outA common mistake
A common mistake in Equations and Inequalities Problem Solving is matching the wrong model to a key phrase. For 'the getaway car held no more than 6 bags,' students often write the equation b = 6 instead of the inequality b ≤ 6, because they see one number and assume it must be exact. The same mistake happens in reverse: for 'the reward split equally among 5 people gave each $40,' students sometimes write the inequality 5p ≥ 40 instead of the equation 5p = 40, because 'split equally' sounds like it could mean a range. Before you submit, ask: does this phrase give one exact value (equation, =) or a limit/range (inequality, <, >, ≤, ≥)?
Lesson 8.7Equations and Inequalities Problem Solving
Version ASupported practice
Learning target I can model and solve real-world problems using equations and inequalities.
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1Circle the letter of the best answer. Show how you know.
Which model is correct for: 'A number divided by 6 equals 7'?
- An / 6 = 7
- B6n = 7
- Cn − 6 = 7
- Dn + 6 = 7
Hint Spot the operation phrase: 'divided by 6'. That phrase gives you the symbol between n and 6.
Show your work -
2Write the name of the correct group on each line.
Sort each problem: equation or inequality?
- Five boxes weigh exactly 60 pounds total. How much does each weigh?
- A number plus 8 equals 20.
- A shelf holds at most 30 books. There are already 17. How many more fit?
- The temperature must stay above 32 degrees.
Hint 'Exactly,' 'equals,' and 'is' signal an equation.
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3Circle the letter of the best answer. Show how you know.
A museum has some paintings. After adding 12, they have 45. Which equation models this?
- Ap + 12 = 45
- Bp − 12 = 45
- C12p = 45
- Dp / 12 = 45
Hint The museum STARTS with an unknown number of paintings p, then 12 more are ADDED to reach 45.
Show your work -
4Circle the letter of the best answer. Show how you know.
A detective needs more than 8 hours to finish the investigation. She has already worked 3 hours. Which inequality represents the additional hours h she needs?
- A3 + h > 8
- B3 + h = 8
- Ch > 3
- D8 + h > 3
Hint There are two time pieces: the 3 hours already worked and the extra hours h. Together they must pass 8 hours.
Show your work -
5Circle the letter of the best answer. Show how you know.
A bus can carry at most 48 passengers. There are already 31 on board. Which inequality shows how many more can board?
- A31 + p ≤ 48
- B31 + p > 48
- C31 + p = 48
- Dp − 31 ≤ 48
Hint The bus already has 31 riders, and p more may board — but the TOTAL cannot pass 48.
Show your work
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6Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake
- 1ProblemA detective already found 8 fingerprints at the scene. After dusting more surfaces, she has 15 fingerprints in all. How many new fingerprints did she find?
- 2Modelx + 8 = 15
- 3Solvex = 15 + 8 = 23
- 4Answer23 new fingerprints.
Which step has the mistake? Explain what went wrong, then show the correct work.
Hint x is increased by 8. What operation undoes adding 8 — adding 8 again, or subtracting 8?
Correct workCorrect answer:
Explain your thinkingHow do you decide whether a word problem needs an equation or an inequality?
Sentence starter I use an equation when the problem says ___, but I use an inequality when the problem says ___ because ___.
Lesson 8.7Equations and Inequalities Problem Solving
Version BCore practice
Learning target I can model and solve real-world problems using equations and inequalities.
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1Complete the table. Show how you found each value.
Write a model, solve, and check reasonableness for each problem.
Word Problem Model Solution Reasonable? 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? Show your work -
2Write the letter of the matching item on each line.
Match each situation to its correct model
- Item 1
- Item 2
- Item 3
- Item 4
- A
- B
- C
- D
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3Decide whether each pair balances. Check Yes or No, then explain.
Solve this two-step problem: 3x + 5 = 26
Left side Right side Balanced? Explain your choice
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4Circle the letter of the best answer. Show how you know.
A student solved 4n = 52 and got n = 13. Is the answer reasonable if n represents the number of notebooks in a box?
- ANo — 13 is too many notebooks
- BNo — the answer should be 48
- CYes — 13 notebooks per box is reasonable
- DYes — but only if n is a fraction
Show your work
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5Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake
- 1ProblemSeven evidence lockers hold 84 case files in all, split equally. How many files per locker?
- 2Model7f = 84
- 3Solvef = 84 − 7 = 77
- 4Answer77 files per locker.
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer:
Explain your thinkingHow do you decide whether a word problem needs an equation or an inequality?
Lesson 8.7Equations and Inequalities Problem Solving
ChallengeExtension
Learning target I can model and solve real-world problems using equations and inequalities.
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1Complete the table. Show how you found each value.
Check each student's model. If wrong, write the correct model and solve.
Problem Student's Model Correct? Fix (if needed) A tank holds at most 500 gallons. It has 320. How many more gallons can be added? 320 + g = 500 Six friends split a pizza bill equally. Each paid $8. What was the total? t / 6 = 8 A runner must complete more than 5 laps. She has done 2. How many more does she need? 2 + m = 5 Show your work
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2Find the mistake. Explain it, then show the correct work.
Find the Problem-Solving Error
- 1ProblemA parking lot holds at most 200 cars. There are 134 already. How many more can park?
- 2Model134 + c = 200
- 3Solvec = 66
- 4AnswerExactly 66 more cars can park.
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer: -
3Answer in complete sentences.
Create two word problems about the same situation: one that requires an equation and one that requires an inequality. Solve both and explain why the models are different.
Write your answer