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

Lesson 8.7Equations and Inequalities Problem Solving

Set B · Version ASecond form · supported practice

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?

RememberEquations and Inequalities Problem Solving

Modeling Real-World Scenarios with Equations & Inequalities 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.

  1. Define the variable and units in the scenario.
  2. Look for keywords: 'equals/is' (=) vs 'at least/limit' (≥, ≤, >, <).
  3. Write the algebraic model, solve with inverse operations, and check in context

Words: Model (Modelar)Equation (Ecuación)Inequality (Desigualdad)Reasonableness (Razonabilidad)Variable (Variable)

  1. 1Circle the letter of the best answer. Show how you know.

    Solve: x − 4 < 11

    1. Ax < 7
    2. Bx > 15
    3. Cx = 15
    4. Dx < 15
    Show your work
  2. 2Circle the letter of the best answer. Show how you know.

    Solve: x + 6 > 14

    1. Ax < 8
    2. Bx = 8
    3. Cx > 8
    4. Dx > 20
    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    Solve: x + 10 ≥ 25

    1. Ax ≥ 15
    2. Bx ≤ 15
    3. Cx = 15
    4. Dx ≥ 35
    Show your work
  4. 4Circle the letter of the best answer. Show how you know.

    Solve: x − 9 ≤ 3

    1. Ax ≤ 6
    2. Bx ≥ 12
    3. Cx ≤ 12
    4. Dx = 12
    Show your work
  5. 5Circle the letter of the best answer. Show how you know.

    A detective has a $500 budget for equipment. She has already spent $320. The inequality 320 + r ≤ 500 represents the remaining amount r she can spend. She also needs exactly 4 flashlights at $15 each, represented by the equation 4f = 60. Solve 320 + r ≤ 500.

    1. Ar ≤ 180
    2. Br ≥ 180
    3. Cr ≤ 500
    4. Dr ≤ 820
    Show your work
  6. 6Circle the letter of the best answer. Show how you know.

    A detective has a $500 budget for equipment. She has already spent $320. The inequality 320 + r ≤ 500 represents the remaining amount r she can spend. She also needs exactly 4 flashlights at $15 each, represented by the equation 4f = 60. Solve 4f = 60.

    1. Af = 15
    2. Bf = 56
    3. Cf = 64
    4. Df = 240
    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
Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.8

Lesson 8.7Equations and Inequalities Problem Solving

Set B · Version BSecond form · core practice

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?

Part 1Understand the ideamodels and meaning
  1. 1Write the name of the correct group on each line.

    Room 1: The case board is a mess. Sort each clue into whether it needs an EQUATION (one exact total) or an INEQUALITY (a limit or range).

    Groups: Needs an Equation (=) Needs an Inequality (<, >, ≤, ≥)
    • The ransom was split equally among 5 suspects, giving each $40.
    • A number of fingerprints plus 9 equals 21.
    • The getaway car held no more than 6 bags of loot.
    • The alarm must stay on for at least 30 minutes.
    • The witness saw fewer than 4 accomplices.
    • Three identical safes hold exactly 75 coins in all.
  2. 2Write the letter of the matching item on each line.

    Room 5: Match each crime-scene clue to the model the detective should write.

    1. The reward divided among 4 detectives gave each $50.
    2. A number of clues take away 7 leaves exactly 9.
    3. The patrol must last more than 8 hours.
    4. The locker holds at least 25 files.
    • A4d = 200, so d = 50
    • Bc − 7 = 9, so c = 16
    • Ch > 8
    • Df ≥ 25
Part 2Apply itreal-world problems
  1. 3Circle the letter of the best answer. Show how you know.

    A detective has a $500 budget for equipment. She has already spent $320. The inequality 320 + r ≤ 500 represents the remaining amount r she can spend. She also needs exactly 4 flashlights at $15 each, represented by the equation 4f = 60. Why is one an equation and the other an inequality?

    1. ABecause equations are for money and inequalities are for counting
    2. BBecause 4 is smaller than 320
    3. CThere is no real difference
    4. DThe flashlights cost exactly $60, but the budget is a limit she must stay under
    Show your work
  2. 4Circle the letter of the best answer. Show how you know.

    Room 2: The vault keypad clue reads, 'A number of stolen coins plus 6 equals 14.' Solve x + 6 = 14 to find the code x.

    1. Ax = 2
    2. Bx = 6
    3. Cx = 8
    4. Dx = 20
    Show your work
  3. 5Circle the letter of the best answer. Show how you know.

    Room 3: A sign warns, 'The safe holds at most 20 gold bars.' Which inequality models the number of bars x?

    1. Ax ≤ 20
    2. Bx ≥ 20
    3. Cx > 20
    4. Dx = 20
    Show your work
  4. 6Circle the letter of the best answer. Show how you know.

    Room 4: A locked drawer clue reads, 'Five identical evidence boxes weigh 60 pounds in all.' Solve 5w = 60 for each box's weight w.

    1. Aw = 12
    2. Bw = 55
    3. Cw = 65
    4. Dw = 300
    Show your work

Explain your thinkingHow do you decide whether a word problem needs an equation or an inequality?

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