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.
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.
The new lesson depends on this prerequisite as active knowledge.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
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 one example of solve multi step equation and inequality problems in the Math Box.
- Change one input, predict the effect, and test the prediction.
- State the relationship or invariant that explains what stayed mathematically consistent.
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.
Solve 3(x + 5) = 24.
x + 5 = 8; x = 3Dividing first or distributing first are equivalent valid strategies.
Solve 4(x + 7) = 44.
x + 7 = 11; x = 4Dividing first or distributing first are equivalent valid strategies.
Solve 5(x + 9) = 70.
x + 9 = 14; x = 5Dividing first or distributing first are equivalent valid strategies.
Solve 6(x + 3) = 54.
x + 3 = 9; x = 6Dividing first or distributing first are equivalent valid strategies.
Solve 7(x + 5) = 84.
x + 5 = 12; x = 7Dividing first or distributing first are equivalent valid strategies.
Solve 8(x + 7) = 120.
x + 7 = 15; x = 8Dividing first or distributing first are equivalent valid strategies.
Solve 2(x + 9) = 22.
x + 9 = 11; x = 2Dividing 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.
Solve 4(x + 5) = 36.
x + 5 = 9; x = 4Dividing first or distributing first are equivalent valid strategies.
Solve 5(x + 7) = 60.
x + 7 = 12; x = 5Dividing first or distributing first are equivalent valid strategies.
Solve 6(x + 9) = 90.
x + 9 = 15; x = 6Dividing first or distributing first are equivalent valid strategies.
Solve 7(x + 3) = 70.
x + 3 = 10; x = 7Dividing first or distributing first are equivalent valid strategies.
Solve 8(x + 5) = 104.
x + 5 = 13; x = 8Dividing first or distributing first are equivalent valid strategies.
Solve 2(x + 7) = 18.
x + 7 = 9; x = 2Dividing first or distributing first are equivalent valid strategies.
Solve 3(x + 9) = 36.
x + 9 = 12; x = 3Dividing 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.
Solve 5(x + 5) = 50.
x + 5 = 10; x = 5Dividing first or distributing first are equivalent valid strategies.
Solve 6(x + 7) = 78.
x + 7 = 13; x = 6Dividing first or distributing first are equivalent valid strategies.
Solve 7(x + 9) = 112.
x + 9 = 16; x = 7Dividing first or distributing first are equivalent valid strategies.
Practice
Guided practice with hints
01Solve 8(x + 3) = 88.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: x + 3 = 11; x = 8
02Solve 2(x + 5) = 14.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: x + 5 = 7; x = 2
03Solve 3(x + 7) = 30.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: x + 7 = 10; x = 3
04Solve 4(x + 9) = 52.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: x + 9 = 13; x = 4
05Solve 5(x + 3) = 40.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: x + 3 = 8; x = 5
06Solve 6(x + 5) = 66.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: x + 5 = 11; x = 6
Practice
Independent practice
01Solve 7(x + 7) = 98.
Answer: x + 7 = 14; x = 7
02Solve 8(x + 9) = 136.
Answer: x + 9 = 17; x = 8
03Solve 2(x + 3) = 10.
Answer: x + 3 = 5; x = 2
04Solve 3(x + 5) = 24.
Answer: x + 5 = 8; x = 3
05Solve 4(x + 7) = 44.
Answer: x + 7 = 11; x = 4
06Solve 5(x + 9) = 70.
Answer: x + 9 = 14; x = 5
07Solve 6(x + 3) = 54.
Answer: x + 3 = 9; x = 6
08Solve 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
Solve 8(x + 7) = 120.
x + 7 = 15; x = 8
Solve 2(x + 9) = 22.
x + 9 = 11; x = 2
Solve 3(x + 3) = 18.
x + 3 = 6; x = 3
Solve 4(x + 5) = 36.
x + 5 = 9; x = 4
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve solve multi-step equation and inequality problems problems accurately and justify each result with a valid check.
Practice
Mastery check
01Solve 5(x + 7) = 60.
Answer: x + 7 = 12; x = 5
02Solve 6(x + 9) = 90.
Answer: x + 9 = 15; x = 6
03Solve 7(x + 3) = 70.
Answer: x + 3 = 10; x = 7
04Solve 8(x + 5) = 104.
Answer: x + 5 = 13; x = 8
05Solve 2(x + 7) = 18.
Answer: x + 7 = 9; x = 2
06Solve 3(x + 9) = 36.
Answer: x + 9 = 12; x = 3
07Solve 4(x + 3) = 28.
Answer: x + 3 = 7; x = 4
08Solve 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
California Mathematics Framework (2023)
Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.Common Core State Standards for Mathematics — Grade 7
Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.