Introduction
Graphing Linear Inequalities in Two Variables is an important Grade 8 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 linear inequalities in two variables.
What Is Graphing Linear Inequalities in Two Variables?
Graphing Linear Inequalities in Two Variables 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 Linear Inequalities in Two Variables
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: The graph shows a solid line and shading above. The point \((3,5)\) is marked in the shaded region. Which inequality is represented?
- A. \(y > x + 1\)
- B. \(y \geq x + 1\)
- C. \(y < x + 1\)
- D. \(y \leq x + 1\)
Why it works: The solid line (boundary included) with shading ABOVE indicates \(y \geq x + 1\). Verify: \((3,5)\) gives \(5 \geq 3+1=4\) ✓ TRUE.
Answer: \(y \geq x + 1\)
Visual Model 2
Question: Based on the dashed line and shaded region, what inequality is shown?
- A. \(y > -x + 2\)
- B. \(y < -x + 2\)
- C. \(y \geq -x + 2\)
- D. \(y \leq -x + 2\)
Why it works: DASHED line means \(>\) or \(<\). Shading BELOW means \(y <\). Combined: \(y < -x + 2\).
Answer: \(y < -x + 2\)
Worked Examples
Example 1
Question: Graph \(y \geq \frac{1}{2}x\). Which region should be shaded?
- A. Below the line only
- B. Above the line only
- C. Above and on the line
- D. Below and on the line
- The inequality \(\geq\) means "greater than or equal," so shade ABOVE the line.
- The SOLID line is included in the solution.
Answer: Above and on the line
Example 2
Question: The graph shows which inequality?
- A. \(y > 2x\)
- B. \(y < 2x\)
- C. \(y \geq 2x\)
- D. \(y \leq 2x\)
- SOLID line (includes boundary) and shading BELOW the line indicates \(y \leq 2x\).
Answer: \(y \leq 2x\)
Example 3
Question: What inequality matches this graph?
- A. \(y > -3x\)
- B. \(y < -3x\)
- C. \(y \geq -3x\)
- D. \(y \leq -3x\)
- The DASHED line and shading ABOVE indicate a strict inequality with \(y\) greater: \(y > -3x\).
Answer: \(y > -3x\)
Real-World Word Problems
Problem 1
Question: A gardener's budget is \(B = 50x + 75y\), where \(x\) is the number of seed packets and \(y\) is the number of tools. The gardener can spend at most $300. Which inequality represents this constraint?
- A. \(50x + 75y \leq 300\)
- B. \(50x + 75y \geq 300\)
- C. \(50x + 75y < 300\)
- D. \(50x + 75y > 300\)
Why it works: At most $300 means the budget is less than or equal to 300. The solid boundary line includes the exact limit of $300.
Answer: \(50x + 75y \leq 300\)
Problem 2
Question: A student graphs \(y \geq x\) and mistakenly shades BELOW the line. What is the error?
- A. Should have used a dashed line
- B. Should have shaded ABOVE the line
- C. Should have shaded BELOW the line
- D. Should have used the inequality \(y < x\)
Why it works: For \(y \geq x\), the solution region is above the line. The boundary is solid because the inequality includes equality.
Answer: Should have shaded ABOVE the line
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 graph represents \(y<2x+1\)?
- A. A dashed line \(y=2x+1\) with shading above
- B. A solid line \(y=2x+1\) with shading below
- C. A dashed line \(y=2x+1\) with shading below
- D. A solid line \(y=2x+1\) with shading above
Question 2
Which inequality has a solid boundary line?
- A. \(y > 3x - 2\)
- B. \(y < x + 5\)
- C. \(y \geq -x + 4\)
- D. \(y \neq 2x - 1\)
Question 3
For the inequality \(y \leq -\frac{1}{2}x + 3\), which point is in the solution region?
- A. \((0, 5)\)
- B. \((0, 0)\)
- C. \((2, 4)\)
- D. \((4, 4)\)
Question 4
For the inequality \(2x + y > 6\), rearrange to slope-intercept form, identify the boundary line type, and determine the shading region.
- A. \(y > -2x + 6\); solid line; shade above
- B. \(y < -2x + 6\); dashed line; shade below
- C. \(y > -2x + 6\); dashed line; shade above
- D. \(y > 2x - 6\); dashed line; shade below
Question 5
Is the point \((2, 4)\) in the solution region of \(y \leq x + 1\)?
- A. Yes, because \(4 \leq 2 + 1\)
- B. Yes, because \(4 < 2 + 1\)
- C. No, because \(4 > 3\)
- D. No, because \(4 = 3\)
Question 6
Which point is on the boundary line of \(3x - y = 9\) but NOT in the solution region of \(3x - y < 9\)?
- A. \((3, 0)\)
- B. \((0, 9)\)
- C. \((1, 6)\)
- D. \((2, 1)\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: A dashed line \(y=2x+1\) with shading below
Strict inequality (\(<\)) means a DASHED boundary line. Because \(y\) is LESS than the expression, shade BELOW the line.
Question 2
Answer: Inequalities with \(\geq\) or \(\leq\) symbols use a solid line.
The symbols \(>\) and \(<\) (strict inequalities) use DASHED lines. The symbols \(\geq\) and \(\leq\) (non-strict) use SOLID lines because the boundary points are INCLUDED in the solution.
Question 3
Answer: \((0, 0)\) is in the solution region.
Test each point in \(y \leq -\frac{1}{2}x + 3\): (A) \(5 \leq -\frac{1}{2}(0)+3=3\)? NO. (B) \(0 \leq 3\)? YES. (C) \(4 \leq -\frac{1}{2}(2)+3=2\)? NO. (D) \(4 \leq -\frac{1}{2}(4)+3=1\)? NO.
Question 4
Answer: \(y > -2x + 6\); dashed line; shade above
Rearrange: \(y > -2x + 6\). Strict inequality (\(>\)) → DASHED boundary. Since \(y\) is GREATER than the expression, shade ABOVE. Distractor A mistakes dashed for solid; B flips both inequality and shading; D has wrong slope.
Question 5
Answer: No, because \(4 > 3\)
Test: \(4 \leq 2 + 1 \Rightarrow 4 \leq 3\) FALSE. So \((2,4)\) is NOT in the solution region.
Question 6
Answer: \((3, 0)\)
Boundary points satisfy \(3x-y=9\). For \((3,0)\), \(3(3)-0=9\), so it is on the boundary. Because the inequality is strict, boundary points are not in the solution region.
Connection to Standards
This lesson supports Grade 8 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 Linear Inequalities in Two Variables 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.
