Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.8

Lesson 8.7Equations and Inequalities Problem Solving

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can model and solve real-world problems using equations and inequalities.

Why This Matters 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?

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
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.'

  1. This has an exact total, so I use an equation. Let t be the total.
  2. Divided among 4 to get 85 means t ÷ 4 = 85, so t = 85 × 4 = 340.
  3. 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?

Answer:

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 out!A 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, <, >, ≤, ≥)?

Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.8

Lesson 8.7Equations and Inequalities Problem Solving

Level 0Most support · every hint printed

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can model and solve real-world problems using equations and inequalities.

Why This Matters 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?

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.'

  1. This has an exact total, so I use an equation. Let t be the total.
  2. Divided among 4 to get 85 means t ÷ 4 = 85, so t = 85 × 4 = 340.
  3. Check reasonableness: $340 split 4 ways is $85 each. That makes sense!
Part 1Understand the ideamodels and meaning
  1. 1Circle the letter of the best answer. Show how you know.

    Which model is correct for: 'A number divided by 6 equals 7'?

    1. An / 6 = 7
    2. B6n = 7
    3. Cn − 6 = 7
    4. Dn + 6 = 7
    Steps to try
    1. Spot the operation phrase: 'divided by 6'. That phrase gives you the symbol between n and 6.
    2. Keep the sentence order — the number n comes first, then the division by 6, then 'equals 7'.
    3. Write n / 6 on the left and 7 on the right. Which choice shows division rather than multiplying or adding 6?
    Show your work
Part 2Apply itreal-world problems
  1. 2Circle 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?

    1. Ap + 12 = 45
    2. Bp − 12 = 45
    3. C12p = 45
    4. Dp / 12 = 45
    Steps to try
    1. The museum STARTS with an unknown number of paintings p, then 12 more are ADDED to reach 45.
    2. 'Adding 12' means the model joins p and 12 with a plus sign, and the total 45 goes after the equals sign.
    3. Write the story in order: p, plus 12, equals 45. Which choice matches without switching the operation?
    Show your work
  2. 3Circle 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?

    1. A3 + h > 8
    2. B3 + h = 8
    3. Ch > 3
    4. D8 + h > 3
    Steps to try
    1. There are two time pieces: the 3 hours already worked and the extra hours h. Together they must pass 8 hours.
    2. 'More than 8' means the TOTAL of 3 + h sits on the big side of a strict inequality — not an equals sign.
    3. Combine first: 3 + h. Then compare that total to 8 using the strict 'more than' symbol. Which choice is built that way?
    Show your work
  3. 4Circle 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?

    1. A31 + p ≤ 48
    2. B31 + p > 48
    3. C31 + p = 48
    4. Dp − 31 ≤ 48
    Steps to try
    1. The bus already has 31 riders, and p more may board — but the TOTAL cannot pass 48.
    2. 'At most 48' allows exactly 48 but nothing above it, so you need the symbol with the 'or equal' bar, not > or =.
    3. Add the parts first: 31 + p. Then cap that sum at 48 with the 'at most' symbol. Which choice matches?
    Show your work

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 ___.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help