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Linear Inequalities in Two Variables

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Written Lesson

An inequality in two variables describes a whole region on a coordinate plane. For example, y ≥ x + 1 does not have just one answer. Many points satisfy it, so we show the answer by shading a region on a graph.

What you will learn

  • Understand inequalities in two variables
  • Draw the boundary line of an inequality
  • Decide whether the boundary line should be solid or broken
  • Use a test point to decide which side to shade
  • Shade regions represented by inequalities such as y ≥ x + 1
  • Recognise inequalities involving horizontal and vertical boundary lines

Key Vocabulary

Term Meaning
Inequality in two variables An inequality involving both x and y, such as y ≥ x + 1.
Boundary line The line that separates the plane into two regions.
Solid line A boundary line used when the line is included in the solution.
Broken line A boundary line used when the line is not included in the solution.
Shaded region The part of the coordinate plane that contains all the solutions.
Test point A point used to check which side of the boundary line satisfies the inequality.

Key Ideas

  • An inequality in two variables is represented by a region on a graph.
  • First draw the boundary line using the matching equation.
  • For ≥ or ≤, use a solid line because the boundary is included.
  • For > or <, use a broken line because the boundary is not included.
  • Use a test point, often (0, 0), to decide which side to shade.
  • If the test point makes the inequality true, shade the side containing the test point.
  • If the test point makes the inequality false, shade the opposite side.

Step 1: Draw the Boundary Line

To draw the boundary line, replace the inequality sign with an equals sign.

Inequality Boundary Line Line Type
y ≥ x + 1 y = x + 1 Solid line
y ≤ 2x - 3 y = 2x - 3 Solid line
y > x + 2 y = x + 2 Broken line
y < -x + 4 y = -x + 4 Broken line

Remember:

  • ≥ means the boundary line is included, so use a solid line.
  • ≤ means the boundary line is included, so use a solid line.
  • > means the boundary line is not included, so use a broken line.
  • < means the boundary line is not included, so use a broken line.

Step 2: Choose a Test Point

After drawing the boundary line, choose a point that is not on the line. The easiest point to test is usually (0, 0).

Important: Use (0, 0) only when it is not on the boundary line. If the boundary passes through the origin, choose another simple point such as (1, 0) or (0, 1).

For example, (0, 0) cannot be used to test y ≤ 2x because it lies on the boundary line y = 2x.

How to use a test point:

  1. Substitute the test point into the inequality.
  2. If the statement is true, shade the side containing the test point.
  3. If the statement is false, shade the other side.

For example, for y ≥ x + 1, test (0, 0):

0 ≥ 0 + 1

0 ≥ 1

This is false, so we do not shade the side containing (0, 0). We shade the opposite side.

Graph of y ≥ x + 1

x y 0 1 2 3 4 5 1 2 3 4 5 (0, 0) false side y = x + 1 shade: y ≥ x + 1 solid line boundary included
The region y ≥ x + 1 uses a solid boundary line and shading above the line.

Solid Line or Broken Line?

The type of inequality sign tells us whether the boundary line is included.

Inequality Sign Boundary Included? Line Type
≥ Yes Solid line
≤ Yes Solid line
> No Broken line
< No Broken line

Worked Examples

Example 1: Show the region y ≥ x + 1.

  1. First draw the boundary line y = x + 1.
  2. Use a solid line because the inequality sign is ≥.
  3. Test the point (0, 0).

0 ≥ 0 + 1

0 ≥ 1

  1. This statement is false.
  2. So shade the side that does not contain (0, 0).
  3. The correct region is above the line.

Answer: Draw a solid line for y = x + 1 and shade above the line.

Example 2: Would y > 2 use a solid or broken line?

  1. The boundary line is y = 2.
  2. The inequality sign is >.
  3. This is a strict inequality, so the boundary is not included.

Answer: Use a broken line at y = 2 and shade above the line.

Example 3: Show the region y < x + 2.

  1. Draw the boundary line y = x + 2.
  2. Use a broken line because the inequality sign is <.
  3. Test (0, 0):

0 < 0 + 2

0 < 2

  1. This is true.
  2. So shade the side containing (0, 0).

Answer: Draw a broken line for y = x + 2 and shade the side containing (0, 0).

Graph of y < x + 2

x y -3-2-1 012 34 -2-11 234 (0, 0) true side y = x + 2 shade: y < x + 2 broken line boundary excluded
The region y < x + 2 uses a broken boundary line. Since (0, 0) makes the inequality true, the side containing the origin is shaded.

Horizontal and Vertical Boundary Lines

Some inequalities make horizontal or vertical boundary lines.

Inequality Boundary Line Region
y > 2 y = 2 Above the horizontal line
y ≤ -1 y = -1 On or below the horizontal line
x ≥ 0 x = 0 On or to the right of the y-axis
x < 3 x = 3 To the left of the vertical line

Important: The inequality x ≥ 0 means all points whose x-coordinates are zero or positive. This is the region on or to the right of the y-axis.

Diagram: x ≥ 0

x y 0 1 2 3 -1 -2 x ≥ 0 shade right of the y-axis
The region x ≥ 0 is on or to the right of the y-axis. The y-axis is included.

Common Mistakes

  • Mistake 1: Using a broken line for ≥ or ≤. These signs need a solid line.
  • Mistake 2: Using a solid line for > or <. These signs need a broken line.
  • Mistake 3: Forgetting to test a point before shading.
  • Mistake 4: Shading below the line automatically. Always check the inequality.
  • Mistake 5: Confusing x ≥ 0 with y ≥ 0. The inequality x ≥ 0 shades right of the y-axis, not above the x-axis.

Multiple Choice Questions

1. For y ≤ x, is the boundary line solid or broken?

  1. Solid
  2. Broken
  3. No boundary line
  4. Vertical only

Answer: A. The sign ≤ means the boundary is included, so the line is solid.

2. For y > x + 2, is the boundary line solid or broken?

  1. Solid
  2. Broken
  3. No line
  4. Horizontal only

Answer: B. The sign > means the boundary is not included, so the line is broken.

3. Which test point is often useful when deciding which side to shade?

  1. (1, 1)
  2. (2, 2)
  3. (0, 0)
  4. (5, 5)

Answer: C. The origin (0, 0) is often the easiest test point.

4. What region is represented by x ≥ 0?

  1. Above the x-axis
  2. Below the x-axis
  3. On or to the right of the y-axis
  4. On or to the left of the y-axis

Answer: C. x ≥ 0 means all points with zero or positive x-coordinates.

Practice Questions

  1. For y ≤ x, is the boundary line solid or broken?
  2. For y > x + 2, is the boundary line solid or broken?
  3. To decide which side to shade, what is a useful test point?
  4. Describe the region represented by x ≥ 0.
  5. For y < 3, is the boundary line solid or broken?
  6. Describe the region represented by y > 2.
  7. For x < -1, is the boundary line vertical or horizontal?
  8. Show the first two steps for graphing y \ge; -x + 4.
  9. Explain why y = x + 1 is the boundary line for y ≥ x + 1.
  10. Explain why a test point helps you decide which side to shade.

Answers

  1. Solid line, because ≤ includes the boundary.
  2. Broken line, because > does not include the boundary.
  3. The point (0, 0), if it is not on the boundary line.
  4. x ≥ 0 represents the region on or to the right of the y-axis.
  5. Broken line, because < does not include the boundary.
  6. y > 2 represents the region above the horizontal line y = 2, using a broken boundary line.
  7. Vertical. The boundary line is x = -1.
  8. First draw the boundary line y = -x + 4. Then use a solid line because the inequality sign is ≥.
  9. The boundary line is found by replacing the inequality sign with an equals sign.
  10. A test point tells you whether that side satisfies the inequality. If the test point makes the inequality true, shade that side. If it is false, shade the opposite side.

Lesson Summary

  • An inequality in two variables is shown as a shaded region on a coordinate plane.
  • The boundary line comes from the matching equation.
  • Use a solid line for ≥ or ≤.
  • Use a broken line for > or <.
  • Use a test point to decide which side to shade.
  • If the test point is true, shade its side.
  • If the test point is false, shade the opposite side.
  • y > or y ≥ usually shades above the boundary line.
  • y < or y ≤ usually shades below the boundary line.

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