Introduction
Graphing Inequalities on a Number Line 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 graphing inequalities on a number line.
What Is Graphing Inequalities on a Number Line?
Graphing Inequalities on a Number Line means reading, creating, and explaining displays so data can answer real questions.
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 Graphing Inequalities on a Number Line
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the title, labels, and scale before answering.
- Use the scale value instead of counting marks as ones when the graph is scaled.
- Compare categories by subtracting or adding values from the display.
- Explain what the data shows in a complete sentence.
Visual Models
Visual Model 1
Question: Which number line represents the inequality \(x>2\)?
- A. An open circle at \(2\) with an arrow pointing right
- B. A closed circle at \(2\) with an arrow pointing right
- C. An open circle at \(2\) with an arrow pointing left
- D. A closed circle at \(2\) with an arrow pointing left
Why it works: \(x>2\) means \(x\) is strictly greater than \(2\), so use an open circle (not including \(2\)) with the arrow pointing toward larger values.
Answer: An open circle at \(2\) with an arrow pointing right
Visual Model 2
Question: Which inequality is represented by the number line?
- A. \(x < 3\)
- B. \(x \le 3\)
- C. \(x \ge 3\)
- D. \(x > 3\)
Why it works: A filled (closed) circle at \(3\) with an arrow pointing right indicates the number \(3\) is included and all numbers greater than \(3\) are solutions.
Answer: \(x \ge 3\)
Worked Examples
Example 1
Question: Which number line shows the solution to \(x \le -1\)?
- A. Closed circle at \(-1\), arrow pointing right
- B. Open circle at \(-1\), arrow pointing left
- C. Closed circle at \(-1\), arrow pointing left
- D. Closed circle at \(1\), arrow pointing left
- \(x \le -1\) means \(x\) equals \(-1\) or is less than \(-1\).
- A closed circle (filled) includes \(-1\), and the arrow points left (toward smaller values).
Answer: Closed circle at \(-1\), arrow pointing left
Example 2
Question: What inequality matches this number line?
- A. \(x > 6\)
- B. \(x < 6\)
- C. \(x \ge 6\)
- D. \(x \le 6\)
- An open circle at \(6\) means \(6\) is not included.
- The arrow points left toward smaller values, so the inequality is \(x < 6\).
Answer: \(x < 6\)
Example 3
Question: Graph the inequality \(x \ge -2\) on a number line. Which description matches your graph?
- A. Closed circle at \(-2\), arrow left
- B. Open circle at \(-2\), arrow right
- C. Closed circle at \(-2\), arrow right
- D. Open circle at \(-2\), arrow left
- \(x \ge -2\) includes \(-2\) (closed circle) and all numbers greater than \(-2\) (arrow points right).
Answer: Closed circle at \(-2\), arrow right
Real-World Word Problems
Problem 1
Question: A student needs to score more than \(80\%\) to pass. Which inequality represents passing scores?
- A. \(s \le 80\)
- B. \(s < 80\)
- C. \(s > 80\)
- D. \(s \ge 80\)
Why it works: "More than \(80\%\)" means strictly greater than \(80\), so the inequality is \(s > 80\) (not \(\ge\)).
Answer: \(s > 80\)
Problem 2
Question: A height requirement states that riders must be taller than \(48\) inches. Which inequality matches this?
- A. \(h \ge 48\)
- B. \(h \le 48\)
- C. \(h < 48\)
- D. \(h > 48\)
Why it works: "Taller than" (strictly greater) means \(h > 48\), not \(h \ge 48\).
Answer: \(h > 48\)
Common Mistakes
- Ignoring the graph scale.
- Reading the wrong category or axis label.
- Answering a comparison question without subtracting.
- Writing a number without explaining what it represents.
Strategy Tips
- Circle the scale before using the graph.
- Write down the value for each category you compare.
- Use addition for totals and subtraction for differences.
- Answer in words so the data result has meaning.
Practice Questions
Question 1
Which inequality represents "all numbers less than \(5\)"?
- A. \(x \le 5\)
- B. \(x < 5\)
- C. \(x > 5\)
- D. \(x \ge 5\)
Question 2
Which number line shows \(x < -3\)?
- A. Open circle at \(-3\), arrow left
- B. Closed circle at \(-3\), arrow left
- C. Open circle at \(-3\), arrow right
- D. Closed circle at \(3\), arrow left
Question 3
A temperature must be at most \(32°\text{F}\) to freeze. Which inequality represents this?
- A. \(T > 32\)
- B. \(T \ge 32\)
- C. \(T < 32\)
- D. \(T \le 32\)
Question 4
Which number line represents \(x \le 0\)?
- A. Open circle at \(0\), arrow left
- B. Closed circle at \(0\), arrow right
- C. Open circle at \(0\), arrow right
- D. Closed circle at \(0\), arrow left
Question 5
Which inequality means "all numbers greater than or equal to \(-5\)"?
- A. \(x > -5\)
- B. \(x < -5\)
- C. \(x \le -5\)
- D. \(x \ge -5\)
Question 6
Which graph shows the solution to \(x \ge 4\)?
- A. Open circle at \(4\), arrow right
- B. Closed circle at \(4\), arrow right
- C. Open circle at \(4\), arrow left
- D. Closed circle at \(4\), arrow left
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(x < 5\)
"Less than" does not include the boundary value, so we use the strict inequality \(x < 5\) (not \(\le\)).
Question 2
Answer: Open circle at \(-3\), arrow left
\(x < -3\) means \(-3\) is not included (open circle) and we want all numbers less than \(-3\) (arrow pointing left).
Question 3
Answer: \(T \le 32\)
"At most" means "less than or equal to," so the inequality is \(T \le 32\).
Question 4
Answer: Closed circle at \(0\), arrow left
\(x \le 0\) includes \(0\) (closed circle) and all numbers less than \(0\) (arrow points left).
Question 5
Answer: \(x \ge -5\)
"Greater than or equal to" uses the \(\ge\) symbol, so the inequality is \(x \ge -5\).
Question 6
Answer: Closed circle at \(4\), arrow right
\(x \ge 4\) includes \(4\) (closed circle) and all numbers greater than \(4\) (arrow points right).
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
Graphing Inequalities on a Number Line becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Read the scale before reading the answer.
