Introduction
Writing Inequalities is an important Grade 6 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with writing inequalities.
What Is Writing Inequalities?
Writing Inequalities means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Writing Inequalities
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: You can practice your instrument for more than \(30\) minutes but must stop before \(60\) minutes. If \(t\) is the time in minutes, which two inequalities best describe this?
| Inequality | Meaning |
|---|---|
| \(t>30\) | more than \(30\) min |
| \(t<60\) | less than \(60\) min |
- A. \(t<30\) and \(t<60\)
- B. \(t>30\) and \(t>60\)
- C. \(t>30\) and \(t<60\)
- D. \(t\leq30\) and \(t\geq60\)
Why it works: "More than \(30\)" is \(t>30\). "Stop before \(60\)" is \(t<60\). Together: \(t>30\) and \(t<60\).
Answer: \(t>30\) and \(t<60\)
Visual Model 2
Question: A baker needs more than \(2\) cups of sugar but no more than \(4\) cups. Which pair of inequalities matches this, using \(s\) for cups?
| Condition | Meaning |
|---|---|
| More than \(2\) cups | \(s>2\) |
| No more than \(4\) cups | \(s\leq4\) |
- A. \(s>2\) and \(s\leq4\)
- B. \(s\geq2\) and \(s<4\)
- C. \(s<2\) and \(s>4\)
- D. \(s\leq2\) and \(s\geq4\)
Why it works: "More than \(2\)" gives \(s>2\). "No more than \(4\)" gives \(s\leq4\). Together: \(s>2\) and \(s\leq4\).
Answer: \(s>2\) and \(s\leq4\)
Worked Examples
Example 1
Question: Which inequality is shown by the graph above?
- A. \(2\leq x<6\)
- B. \(2
- C. \(2
- D. \(2\leq x\leq6\)
- C. \(2
- The filled circle at \(2\) means "equals \(2\)" (\(\geq\)).
- The open circle at \(6\) means "not including \(6\)" (\(<\)).
- Together: \(2\leq x<6\).
Answer: \(2\leq x<6\)
Example 2
Question: Based on the number line above, which inequality is represented?
- A. \(0
- B. \(0\leq x<5\)
- C. \(0
- D. \(0\leq x\leq5\)
- The open circle at \(0\) means "not including \(0\)" (\(>\)).
- The filled circle at \(5\) means "including \(5\)" (\(\leq\)).
- Together: \(0<x\leq5\).
Answer: \(0
Example 3
Question: Which inequality matches the number line above?
- A. \(-4\leq x<-1\)
- B. \(-4
- C. \(-4
- D. \(-4\leq x\leq-1\)
- C. \(-4
- The filled circle at \(-4\) means "including \(-4\)" (\(\geq\)).
- The open circle at \(-1\) means "not including \(-1\)" (\(<\)).
- Together: \(-4\leq x<-1\).
Answer: \(-4\leq x<-1\)
Real-World Word Problems
Problem 1
Question: A roller coaster requires riders to be at least \(48\) inches tall. Which inequality describes the height \(h\) (in inches) needed to ride?
- A. \(h<48\)
- B. \(h>48\)
- C. \(h\leq48\)
- D. \(h\geq48\)
Why it works: "At least \(48\)" means \(48\) or more, which is written as \(h\geq48\).
Answer: \(h\geq48\)
Problem 2
Question: A recipe calls for no more than \(\frac{1}{2}\) cup of salt. Which inequality represents the amount \(a\) of salt that can be used?
- A. \(a<\frac{1}{2}\)
- B. \(a\geq\frac{1}{2}\)
- C. \(a\leq\frac{1}{2}\)
- D. \(a>\frac{1}{2}\)
Why it works: "No more than" means at most: \(a\leq\frac{1}{2}\).
Answer: \(a\leq\frac{1}{2}\)
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
The speed limit on the highway is no more than \(65\) miles per hour. Which inequality represents the allowed speed \(s\)?
- A. \(s>65\)
- B. \(s\leq65\)
- C. \(s\geq65\)
- D. \(s=65\)
Question 2
A swimming pool must have a temperature greater than \(78\) degrees Fahrenheit. Which inequality matches this requirement for temperature \(t\)?
- A. \(t\leq78\)
- B. \(t=78\)
- C. \(t>78\)
- D. \(t<78\)
Question 3
To qualify for a school club, you must have no less than a \(3.0\) grade point average. Which inequality describes the minimum GPA \(g\)?
- A. \(g<3.0\)
- B. \(g>3.0\)
- C. \(g\leq3.0\)
- D. \(g\geq3.0\)
Question 4
A discount applies when you spend less than \($50\). If \(d\) represents the amount you spend, which inequality is correct?
- A. \(d<50\)
- B. \(d\leq50\)
- C. \(d>50\)
- D. \(d\geq50\)
Question 5
Which phrase best describes the inequality \(x\geq12\)?
- A. Less than \(12\)
- B. Greater than \(12\)
- C. No more than \(12\)
- D. At least \(12\)
Question 6
The weight of a piece of luggage must not exceed \(50\) pounds. Which inequality best represents this, where \(w\) is the weight?
- A. \(w>50\)
- B. \(w\leq50\)
- C. \(w<50\)
- D. \(w\geq50\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(s\leq65\)
"No more than" means at most \(65\), which allows \(65\) and anything less: \(s\leq65\).
Question 2
Answer: \(t>78\)
"Greater than" means strictly more, not equal: \(t>78\). This excludes \(78\) itself.
Question 3
Answer: \(g\geq3.0\)
"No less than \(3.0\)" means \(3.0\) or more: \(g\geq3.0\).
Question 4
Answer: \(d<50\)
"Less than \($50\)" means strictly below \(50\): \(d<50\).
Question 5
Answer: At least \(12\)
\(x\geq12\) means \(x\) can equal \(12\) or be larger, which matches "at least \(12\)."
Question 6
Answer: \(w\leq50\)
"Must not exceed" means at most \(50\): \(w\leq50\).
Connection to Standards
This lesson supports Grade 6 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Writing Inequalities becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
