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.
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.
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 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 one example of solve two step inequalities 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 2x + 3 > 9.
2x > 6; x > 3Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 3x + 5 > 17.
3x > 12; x > 4Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 4x + 7 > 27.
4x > 20; x > 5Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 5x + 9 > 39.
5x > 30; x > 6Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 6x + 3 > 45.
6x > 42; x > 7Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 7x + 5 > 61.
7x > 56; x > 8Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 8x + 7 > 79.
8x > 72; x > 9Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 2x + 9 > 29.
2x > 20; x > 10Inverse 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.
Solve 4x + 5 > 53.
4x > 48; x > 12Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 5x + 7 > 72.
5x > 65; x > 13Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 6x + 9 > 93.
6x > 84; x > 14Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 7x + 3 > 108.
7x > 105; x > 15Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 8x + 5 > 133.
8x > 128; x > 16Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 2x + 7 > 41.
2x > 34; x > 17Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 3x + 9 > 63.
3x > 54; x > 18Inverse 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.
Solve 5x + 5 > 105.
5x > 100; x > 20Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 6x + 7 > 133.
6x > 126; x > 21Inverse operations preserve the inequality direction when dividing by a positive number.
Solve 7x + 9 > 163.
7x > 154; x > 22Inverse operations preserve the inequality direction when dividing by a positive number.
Practice
Guided practice with hints
01Solve 8x + 3 > 187.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 8x > 184; x > 23
02Solve 2x + 5 > 53.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 2x > 48; x > 24
03Solve 3x + 7 > 82.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 3x > 75; x > 25
04Solve 4x + 9 > 113.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 4x > 104; x > 26
05Solve 5x + 3 > 138.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 5x > 135; x > 27
06Solve 6x + 5 > 173.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 6x > 168; x > 28
Practice
Independent practice
01Solve 7x + 7 > 210.
Answer: 7x > 203; x > 29
02Solve 8x + 9 > 249.
Answer: 8x > 240; x > 30
03Solve 2x + 3 > 65.
Answer: 2x > 62; x > 31
04Solve 3x + 5 > 101.
Answer: 3x > 96; x > 32
05Solve 4x + 7 > 139.
Answer: 4x > 132; x > 33
06Solve 5x + 9 > 179.
Answer: 5x > 170; x > 34
07Solve 6x + 3 > 213.
Answer: 6x > 210; x > 35
08Solve 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
Solve 8x + 7 > 303.
8x > 296; x > 37
Solve 2x + 9 > 85.
2x > 76; x > 38
Solve 3x + 3 > 120.
3x > 117; x > 39
Solve 4x + 5 > 165.
4x > 160; x > 40
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve solve two-step inequalities problems accurately and justify each result with a valid check.
Practice
Mastery check
01Solve 5x + 7 > 212.
Answer: 5x > 205; x > 41
02Solve 6x + 9 > 261.
Answer: 6x > 252; x > 42
03Solve 7x + 3 > 304.
Answer: 7x > 301; x > 43
04Solve 8x + 5 > 357.
Answer: 8x > 352; x > 44
05Solve 2x + 7 > 97.
Answer: 2x > 90; x > 45
06Solve 3x + 9 > 147.
Answer: 3x > 138; x > 46
07Solve 4x + 3 > 191.
Answer: 4x > 188; x > 47
08Solve 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
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.