Chapter 3 · Lesson 7 of 10 · Grade 7 · Expressions & Equations · MAT-07-EE-007
Solve Two-Step Inequalities
Represent and solve solve two-step inequalities 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 7 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 two-step inequalities problems using precise mathematical notation, models, and reasoning.
- Explain why a method for solve two-step inequalities 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 7 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 Two-Step Inequalities uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.
Represent the relationship
The Math Box uses inequality line 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.
Inequality Graph Lab
- Model solve two step inequalities 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 2x + 3 > 9.
2x > 6; x > 3Inverse operations preserve the inequality direction when dividing by a positive number.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 3x + 5 > 17.
3x > 12; x > 4The representation should show the same mathematical relationship as the calculation. Inverse operations preserve the inequality direction when dividing by a positive number.
Explain why a valid method works, then solve: Solve 4x + 7 > 27.
4x > 20; x > 5A complete explanation names the relationship or property being preserved. Inverse operations preserve the inequality direction when dividing by a positive number.
Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 5x + 9 > 39.
5x > 30; x > 6The estimate is a reasonableness check, not a replacement for the exact result. Inverse operations preserve the inequality direction when dividing by a positive number.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 6x + 3 > 45.
6x > 42; x > 7Verification should independently support the result. Inverse operations preserve the inequality direction when dividing by a positive number.
Interpret the answer in context after solving. What does the result mean here? Solve 7x + 5 > 61.
7x > 56; x > 8State the result with its meaning, units, direction, or comparison—not only a number. Inverse operations preserve the inequality direction when dividing by a positive number.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 8x + 7 > 79.
Use equivalent operations on both sides. Correct solution: 8x > 72; x > 9Error analysis requires identifying the broken idea, not merely replacing the final answer. Inverse operations preserve the inequality direction when dividing by a positive number.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 2x + 9 > 29. Case B: Solve 4x + 3 > 79.
Case A: 2x > 20; x > 10 Case B: 4x > 76; x > 19Compare the structure, representation, units, and result rather than only the numbers. Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 3x + 3 > 36.
3x > 33; x > 11Inverse operations preserve the inequality direction when dividing by a positive number.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 4x + 5 > 53.
4x > 48; x > 12The representation should show the same mathematical relationship as the calculation. Inverse operations preserve the inequality direction when dividing by a positive number.
Explain why a valid method works, then solve: Solve 5x + 7 > 72.
5x > 65; x > 13A complete explanation names the relationship or property being preserved. Inverse operations preserve the inequality direction when dividing by a positive number.
Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 6x + 9 > 93.
6x > 84; x > 14The estimate is a reasonableness check, not a replacement for the exact result. Inverse operations preserve the inequality direction when dividing by a positive number.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 7x + 3 > 108.
7x > 105; x > 15Verification should independently support the result. Inverse operations preserve the inequality direction when dividing by a positive number.
Interpret the answer in context after solving. What does the result mean here? Solve 8x + 5 > 133.
8x > 128; x > 16State the result with its meaning, units, direction, or comparison—not only a number. Inverse operations preserve the inequality direction when dividing by a positive number.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 2x + 7 > 41.
Use equivalent operations on both sides. Correct solution: 2x > 34; x > 17Error analysis requires identifying the broken idea, not merely replacing the final answer. Inverse operations preserve the inequality direction when dividing by a positive number.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 3x + 9 > 63. Case B: Solve 5x + 3 > 138.
Case A: 3x > 54; x > 18 Case B: 5x > 135; x > 27Compare the structure, representation, units, and result rather than only the numbers. Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 4x + 3 > 79.
4x > 76; x > 19Inverse operations preserve the inequality direction when dividing by a positive number.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 5x + 5 > 105.
5x > 100; x > 20The representation should show the same mathematical relationship as the calculation. Inverse operations preserve the inequality direction when dividing by a positive number.
Explain why a valid method works, then solve: Solve 6x + 7 > 133.
6x > 126; x > 21A complete explanation names the relationship or property being preserved. Inverse operations preserve the inequality direction when dividing by a positive number.
Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 7x + 9 > 163.
7x > 154; x > 22The estimate is a reasonableness check, not a replacement for the exact result. Inverse operations preserve the inequality direction when dividing by a positive number.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 8x + 3 > 187.
Hint: Choose a representation before computing.
Answer: 8x > 184; x > 23
02Interpret the answer in context after solving. What does the result mean here? Solve 2x + 5 > 53.
Hint: Explain why the method works and name a verification strategy.
Answer: 2x > 48; x > 24
03Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 3x + 7 > 82.
Hint: Choose a representation before computing.
Answer: Use equivalent operations on both sides. Correct solution: 3x > 75; x > 25
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 4x + 9 > 113. Case B: Solve 6x + 3 > 213.
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: 4x > 104; x > 26 Case B: 6x > 210; x > 35
05Solve 5x + 3 > 138.
Hint: Choose a representation before computing.
Answer: 5x > 135; x > 27
06Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 6x + 5 > 173.
Hint: Explain why the method works and name a verification strategy.
Answer: 6x > 168; x > 28
Practice
Independent practice
01Explain why a valid method works, then solve: Solve 7x + 7 > 210.
Answer: 7x > 203; x > 29
02Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 8x + 9 > 249.
Answer: 8x > 240; x > 30
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 2x + 3 > 65.
Answer: 2x > 62; x > 31
04Interpret the answer in context after solving. What does the result mean here? Solve 3x + 5 > 101.
Answer: 3x > 96; x > 32
05Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Solve 4x + 7 > 139.
Answer: Use equivalent operations on both sides. Correct solution: 4x > 132; x > 33
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 5x + 9 > 179. Case B: Solve 7x + 3 > 304.
Answer: Case A: 5x > 170; x > 34 Case B: 7x > 301; x > 43
07Solve 6x + 3 > 213.
Answer: 6x > 210; x > 35
08Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 7x + 5 > 257.
Answer: 7x > 252; x > 36
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 8x + 7 > 303.
8x > 296; x > 37
Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 2x + 9 > 85.
2x > 76; x > 38
Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 3x + 3 > 120.
3x > 117; x > 39
Interpret the answer in context after solving. What does the result mean here? Solve 4x + 5 > 165.
4x > 160; x > 40
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 3 Reasoning Lab
Solve solve two-step inequalities 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 5x + 7 > 212.
Answer: Use equivalent operations on both sides. Correct solution: 5x > 205; x > 41
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Solve 6x + 9 > 261. Case B: Solve 8x + 3 > 411.
Answer: Case A: 6x > 252; x > 42 Case B: 8x > 408; x > 51
03Solve 7x + 3 > 304.
Answer: 7x > 301; x > 43
04Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Solve 8x + 5 > 357.
Answer: 8x > 352; x > 44
05Explain why a valid method works, then solve: Solve 2x + 7 > 97.
Answer: 2x > 90; x > 45
06Estimate or predict first, then calculate and decide whether the result is reasonable: Solve 3x + 9 > 147.
Answer: 3x > 138; x > 46
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Solve 4x + 3 > 191.
Answer: 4x > 188; x > 47
08Interpret the answer in context after solving. What does the result mean here? Solve 5x + 5 > 245.
Answer: 5x > 240; x > 48
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.