Chapter 3 · Lesson 9 of 10 · Grade 7 · Expressions & Equations · MAT-07-EE-009
Solve Multi-Step Equation and Inequality Problems
Represent and solve solve multi-step equation and inequality problems problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 3: Expressions, Equations, and Inequalities
Expressions & Equations
Essential question: How can equivalent expressions and equations model constraints and support justified solutions?
Where this lesson fits
Lesson 9 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create and defend a multi-step model using the central ideas of expressions, equations, and inequalities, including a representation, calculation, interpretation, and independent check.
Learning objectives
What you should be able to do
- Represent and solve solve multi-step equation and inequality problems problems using precise mathematical notation, models, and reasoning.
- Explain why a method for solve multi-step equation and inequality problems works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
Grade 7 uses prerequisite knowledge as active reasoning.
Grade 7 uses prerequisite knowledge as active reasoning.
Learn
Build the idea from meaning, not memorization.
Chapter 3: Expressions, Equations, and Inequalities
Expressions & Equations. Lesson 9 of 10 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 7 chapters, Grade 6 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier chapter representation.
Mathematical meaning
Solve Multi-Step Equation and Inequality Problems uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.
Represent the relationship
The Math Box uses equation balance to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify algebra by substituting the proposed solution into the original statement; equivalent expressions must agree for the same input.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Equation Balance
Adjust both sides. Equality means both expressions have the same value.
Balanced: both sides have equal value.
- Model solve multi step equation and inequality problems in the Math Box.
- Change one input and predict the effect before testing it.
- Connect the visual, numerical, and symbolic representations.
- State what remains invariant and why.
Worked examples
Twenty different ways to see the concept work.
Solve 2(x + 3) = 10.
x + 3 = 5; x = 2Dividing first or distributing first are equivalent valid strategies.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 3(x + 5) = 24.
x + 5 = 8; x = 3The representation should show the same mathematical relationship as the calculation. Dividing first or distributing first are equivalent valid strategies.
Explain why a valid method works, then solve: Solve 4(x + 7) = 44.
x + 7 = 11; x = 4A complete explanation names the relationship or property being preserved. Dividing first or distributing first are equivalent valid strategies.
Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 5(x + 9) = 70.
x + 9 = 14; x = 5The estimate is a reasonableness check, not a replacement for the exact result. Dividing first or distributing first are equivalent valid strategies.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 6(x + 3) = 54.
x + 3 = 9; x = 6Verification should independently support the result. Dividing first or distributing first are equivalent valid strategies.
Interpret the answer in context after solving. What does the result mean here? Solve 7(x + 5) = 84.
x + 5 = 12; x = 7State the result with its meaning, units, direction, or comparison—not only a number. Dividing first or distributing first are equivalent valid strategies.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 8(x + 7) = 120.
Use equivalent operations on both sides. Correct solution: x + 7 = 15; x = 8Error analysis requires identifying the broken idea, not merely replacing the final answer. Dividing first or distributing first are equivalent valid strategies.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 2(x + 9) = 22. Case B: Solve 4(x + 3) = 28.
Case A: x + 9 = 11; x = 2 Case B: x + 3 = 7; x = 4Compare the structure, representation, units, and result rather than only the numbers. Dividing first or distributing first are equivalent valid strategies.
Solve 3(x + 3) = 18.
x + 3 = 6; x = 3Dividing first or distributing first are equivalent valid strategies.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 4(x + 5) = 36.
x + 5 = 9; x = 4The representation should show the same mathematical relationship as the calculation. Dividing first or distributing first are equivalent valid strategies.
Explain why a valid method works, then solve: Solve 5(x + 7) = 60.
x + 7 = 12; x = 5A complete explanation names the relationship or property being preserved. Dividing first or distributing first are equivalent valid strategies.
Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 6(x + 9) = 90.
x + 9 = 15; x = 6The estimate is a reasonableness check, not a replacement for the exact result. Dividing first or distributing first are equivalent valid strategies.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 7(x + 3) = 70.
x + 3 = 10; x = 7Verification should independently support the result. Dividing first or distributing first are equivalent valid strategies.
Interpret the answer in context after solving. What does the result mean here? Solve 8(x + 5) = 104.
x + 5 = 13; x = 8State the result with its meaning, units, direction, or comparison—not only a number. Dividing first or distributing first are equivalent valid strategies.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 2(x + 7) = 18.
Use equivalent operations on both sides. Correct solution: x + 7 = 9; x = 2Error analysis requires identifying the broken idea, not merely replacing the final answer. Dividing first or distributing first are equivalent valid strategies.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 3(x + 9) = 36. Case B: Solve 5(x + 3) = 40.
Case A: x + 9 = 12; x = 3 Case B: x + 3 = 8; x = 5Compare the structure, representation, units, and result rather than only the numbers. Dividing first or distributing first are equivalent valid strategies.
Solve 4(x + 3) = 28.
x + 3 = 7; x = 4Dividing first or distributing first are equivalent valid strategies.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 5(x + 5) = 50.
x + 5 = 10; x = 5The representation should show the same mathematical relationship as the calculation. Dividing first or distributing first are equivalent valid strategies.
Explain why a valid method works, then solve: Solve 6(x + 7) = 78.
x + 7 = 13; x = 6A complete explanation names the relationship or property being preserved. Dividing first or distributing first are equivalent valid strategies.
Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 7(x + 9) = 112.
x + 9 = 16; x = 7The estimate is a reasonableness check, not a replacement for the exact result. Dividing first or distributing first are equivalent valid strategies.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 8(x + 3) = 88.
Hint: Choose a representation before computing.
Answer: x + 3 = 11; x = 8
02Interpret the answer in context after solving. What does the result mean here? Solve 2(x + 5) = 14.
Hint: Explain why the method works and name a verification strategy.
Answer: x + 5 = 7; x = 2
03Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 3(x + 7) = 30.
Hint: Choose a representation before computing.
Answer: Use equivalent operations on both sides. Correct solution: x + 7 = 10; x = 3
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 4(x + 9) = 52. Case B: Solve 6(x + 3) = 54.
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: x + 9 = 13; x = 4 Case B: x + 3 = 9; x = 6
05Solve 5(x + 3) = 40.
Hint: Choose a representation before computing.
Answer: x + 3 = 8; x = 5
06Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 6(x + 5) = 66.
Hint: Explain why the method works and name a verification strategy.
Answer: x + 5 = 11; x = 6
Practice
Independent practice
01Explain why a valid method works, then solve: Solve 7(x + 7) = 98.
Answer: x + 7 = 14; x = 7
02Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 8(x + 9) = 136.
Answer: x + 9 = 17; x = 8
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 2(x + 3) = 10.
Answer: x + 3 = 5; x = 2
04Interpret the answer in context after solving. What does the result mean here? Solve 3(x + 5) = 24.
Answer: x + 5 = 8; x = 3
05Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 4(x + 7) = 44.
Answer: Use equivalent operations on both sides. Correct solution: x + 7 = 11; x = 4
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 5(x + 9) = 70. Case B: Solve 7(x + 3) = 70.
Answer: Case A: x + 9 = 14; x = 5 Case B: x + 3 = 10; x = 7
07Solve 6(x + 3) = 54.
Answer: x + 3 = 9; x = 6
08Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 7(x + 5) = 84.
Answer: x + 5 = 12; x = 7
Common mistakes
Learn to catch the error, not just the answer.
Use equivalent operations on both sides.
Only terms with the same variable part are like terms.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Translate cost, distance, rate, and measurement situations into symbolic rules.
- Use equations and inequalities to determine unknown quantities, thresholds, and feasible ranges.
Challenge problems
Explain why a valid method works, then solve: Solve 8(x + 7) = 120.
x + 7 = 15; x = 8
Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 2(x + 9) = 22.
x + 9 = 11; x = 2
Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 3(x + 3) = 18.
x + 3 = 6; x = 3
Interpret the answer in context after solving. What does the result mean here? Solve 4(x + 5) = 36.
x + 5 = 9; x = 4
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 3 Reasoning Lab
Solve solve multi-step equation and inequality problems tasks using calculation, representation, explanation, interpretation, verification, error analysis, and comparison.
Practice
Mastery check
01Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 5(x + 7) = 60.
Answer: Use equivalent operations on both sides. Correct solution: x + 7 = 12; x = 5
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 6(x + 9) = 90. Case B: Solve 8(x + 3) = 88.
Answer: Case A: x + 9 = 15; x = 6 Case B: x + 3 = 11; x = 8
03Solve 7(x + 3) = 70.
Answer: x + 3 = 10; x = 7
04Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 8(x + 5) = 104.
Answer: x + 5 = 13; x = 8
05Explain why a valid method works, then solve: Solve 2(x + 7) = 18.
Answer: x + 7 = 9; x = 2
06Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 3(x + 9) = 36.
Answer: x + 9 = 12; x = 3
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 4(x + 3) = 28.
Answer: x + 3 = 7; x = 4
08Interpret the answer in context after solving. What does the result mean here? Solve 5(x + 5) = 50.
Answer: x + 5 = 10; x = 5
Terminology
Words to know
- variable
- A symbol representing a number that may vary or be unknown.
- coefficient
- A numerical factor multiplying a variable.
- expression
- A combination of numbers, variables, and operations.
- solution
- A value that makes an equation or inequality true.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 7 scope, sequence, and public task structure
Coherent sequencing, representations, and reasoning-rich tasks.Eureka Math² Grade 7 program structure
Module coherence, cumulative practice, and fluency.enVision Mathematics Grade 7 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 7
Grade-level content and mathematical-practice expectations.