Chapter 4 · Lesson 9 of 10 · Grade 6 · Expressions & Equations · MAT-06-EE-009
Write and Graph Inequalities
Represent and solve write and graph inequalities problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 4: Expressions, Equations, and Inequalities
Using symbols to describe relationships
Essential question: How can a verbal or numerical relationship be expressed symbolically, transformed without changing its meaning, and solved?
Where this lesson fits
Lesson 9 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Model a real pricing or distance situation with variables, expressions, equations, inequalities, tables, and graphs.
Learning objectives
What you should be able to do
- Represent and solve write and graph inequalities problems using precise mathematical notation, models, and reasoning.
- Explain why a method for write and graph 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.
Learn
Build the idea from meaning, not memorization.
Chapter 4: Expressions, Equations, and Inequalities
Using symbols to describe relationships. This is Lesson 9 of 10 in Chapter 4. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes Chapter 1 rates, Chapter 2 operation fluency, Chapter 3 signed quantities. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Write and Graph 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 write and graph inequalities in the Math Box.
- Change one input, predict the effect, and test the prediction.
- Connect the model to another representation used earlier in this chapter.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
Write an inequality for all numbers greater than 5.
x > 5Use an open endpoint at the boundary and shade right.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Write an inequality for all numbers greater than 6.
x > 6The representation should show the same mathematical relationship as the calculation. Use an open endpoint at the boundary and shade right.
Explain why a valid method works, then solve: Write an inequality for all numbers greater than 7.
x > 7A complete explanation names the relationship or property being preserved. Use an open endpoint at the boundary and shade right.
Estimate or predict first, then calculate and decide whether the result is reasonable: Write an inequality for all numbers greater than 8.
x > 8The estimate is a reasonableness check, not a replacement for the exact result. Use an open endpoint at the boundary and shade right.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Write an inequality for all numbers greater than 9.
x > 9Verification should independently support the result. Use an open endpoint at the boundary and shade right.
Interpret the answer in context after solving. What does the result mean here? Write an inequality for all numbers greater than 10.
x > 10State the result with its meaning, units, direction, or comparison—not only a number. Use an open endpoint at the boundary and shade right.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Write an inequality for all numbers greater than 11.
Use equivalent operations on both sides. Correct solution: x > 11Error analysis requires identifying the broken idea, not merely replacing the final answer. Use an open endpoint at the boundary and shade right.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Write an inequality for all numbers greater than 12. Case B: Write an inequality for all numbers greater than 21.
Case A: x > 12 Case B: x > 21Compare the structure, representation, units, and result rather than only the numbers. Use an open endpoint at the boundary and shade right.
Write an inequality for all numbers greater than 13.
x > 13Use an open endpoint at the boundary and shade right.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Write an inequality for all numbers greater than 14.
x > 14The representation should show the same mathematical relationship as the calculation. Use an open endpoint at the boundary and shade right.
Explain why a valid method works, then solve: Write an inequality for all numbers greater than 15.
x > 15A complete explanation names the relationship or property being preserved. Use an open endpoint at the boundary and shade right.
Estimate or predict first, then calculate and decide whether the result is reasonable: Write an inequality for all numbers greater than 16.
x > 16The estimate is a reasonableness check, not a replacement for the exact result. Use an open endpoint at the boundary and shade right.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Write an inequality for all numbers greater than 17.
x > 17Verification should independently support the result. Use an open endpoint at the boundary and shade right.
Interpret the answer in context after solving. What does the result mean here? Write an inequality for all numbers greater than 18.
x > 18State the result with its meaning, units, direction, or comparison—not only a number. Use an open endpoint at the boundary and shade right.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Write an inequality for all numbers greater than 19.
Use equivalent operations on both sides. Correct solution: x > 19Error analysis requires identifying the broken idea, not merely replacing the final answer. Use an open endpoint at the boundary and shade right.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Write an inequality for all numbers greater than 20. Case B: Write an inequality for all numbers greater than 29.
Case A: x > 20 Case B: x > 29Compare the structure, representation, units, and result rather than only the numbers. Use an open endpoint at the boundary and shade right.
Write an inequality for all numbers greater than 21.
x > 21Use an open endpoint at the boundary and shade right.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Write an inequality for all numbers greater than 22.
x > 22The representation should show the same mathematical relationship as the calculation. Use an open endpoint at the boundary and shade right.
Explain why a valid method works, then solve: Write an inequality for all numbers greater than 23.
x > 23A complete explanation names the relationship or property being preserved. Use an open endpoint at the boundary and shade right.
Estimate or predict first, then calculate and decide whether the result is reasonable: Write an inequality for all numbers greater than 24.
x > 24The estimate is a reasonableness check, not a replacement for the exact result. Use an open endpoint at the boundary and shade right.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Write an inequality for all numbers greater than 25.
Hint: Choose a representation before computing.
Answer: x > 25
02Interpret the answer in context after solving. What does the result mean here? Write an inequality for all numbers greater than 26.
Hint: Explain what relationship or property makes your method valid.
Answer: x > 26
03Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Write an inequality for all numbers greater than 27.
Hint: Choose a representation before computing.
Answer: Use equivalent operations on both sides. Correct solution: x > 27
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Write an inequality for all numbers greater than 28. Case B: Write an inequality for all numbers greater than 37.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: x > 28 Case B: x > 37
05Write an inequality for all numbers greater than 29.
Hint: Choose a representation before computing.
Answer: x > 29
06Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Write an inequality for all numbers greater than 30.
Hint: Explain what relationship or property makes your method valid.
Answer: x > 30
Practice
Independent practice
01Explain why a valid method works, then solve: Write an inequality for all numbers greater than 31.
Answer: x > 31
02Estimate or predict first, then calculate and decide whether the result is reasonable: Write an inequality for all numbers greater than 32.
Answer: x > 32
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Write an inequality for all numbers greater than 33.
Answer: x > 33
04Interpret the answer in context after solving. What does the result mean here? Write an inequality for all numbers greater than 34.
Answer: x > 34
05Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Write an inequality for all numbers greater than 35.
Answer: Use equivalent operations on both sides. Correct solution: x > 35
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Write an inequality for all numbers greater than 36. Case B: Write an inequality for all numbers greater than 45.
Answer: Case A: x > 36 Case B: x > 45
07Write an inequality for all numbers greater than 37.
Answer: x > 37
08Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Write an inequality for all numbers greater than 38.
Answer: x > 38
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: Write an inequality for all numbers greater than 39.
x > 39
Estimate or predict first, then calculate and decide whether the result is reasonable: Write an inequality for all numbers greater than 40.
x > 40
Solve and verify the result with a second method, inverse operation, or equivalent representation: Write an inequality for all numbers greater than 41.
x > 41
Interpret the answer in context after solving. What does the result mean here? Write an inequality for all numbers greater than 42.
x > 42
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 4 Reasoning Lab
Solve write and graph inequalities problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Write an inequality for all numbers greater than 43.
Answer: Use equivalent operations on both sides. Correct solution: x > 43
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Write an inequality for all numbers greater than 44. Case B: Write an inequality for all numbers greater than 53.
Answer: Case A: x > 44 Case B: x > 53
03Write an inequality for all numbers greater than 45.
Answer: x > 45
04Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Write an inequality for all numbers greater than 46.
Answer: x > 46
05Explain why a valid method works, then solve: Write an inequality for all numbers greater than 47.
Answer: x > 47
06Estimate or predict first, then calculate and decide whether the result is reasonable: Write an inequality for all numbers greater than 48.
Answer: x > 48
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Write an inequality for all numbers greater than 49.
Answer: x > 49
08Interpret the answer in context after solving. What does the result mean here? Write an inequality for all numbers greater than 50.
Answer: x > 50
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 6 scope, sequence, standards, and public task structure
Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.Eureka Math² Grade 6 program structure
Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.enVision Mathematics Grade 6 instructional model
Reference for problem-based entry points, visual learning, modeling, and application patterns.Common Core State Standards for Mathematics — Grade 6
Grade-level content and mathematical-practice expectations.