0580

Solving and Graphing Inequalities

Coordinate Geometry and Graphs

Which line, and how to draw it

  • Replace the inequality sign with an equals sign and draw that line first
  • Draw a solid line when the sign is ⩽ or ⩾, because the points along it are included
  • Draw a broken line when the sign is < or >, because the points along it are excluded
  • Putting a sloping boundary into y = mx + c makes it far easier to plot
    • 2x + y ⩽ 8 has boundary 2x + y = 8, which is y = 8 − 2x, meeting the axes at (0, 8) and (4, 0)
  • Extend every line right across the grid you are given, not just as far as the region looks like it goes
Three boundary lines drawn across a grid before any shading: 3x + 2y = 12 solid and sloping down, y = 2x broken and sloping up, and x = 3 broken and vertical
Source: Finding regions using inequalities by Save My Exams
  • Each boundary line drawn correctly is worth a mark before any shading happens
Worked example

Choosing the boundary lines

A region is defined by y ⩽ 2x + 1, x > 3 and y ⩾ 0. Describe the three boundary lines you would draw.

Solution:

  • Replace each inequality sign with an equals sign to get the boundary
  • y ⩽ 2x + 1 gives the line y = 2x + 1, a sloping line through (0, 1) with gradient 2
  • The sign is ⩽, which includes the boundary, so that line is drawn solid
  • x > 3 gives the line x = 3, a vertical line through 3 on the x-axis
  • The sign is >, which excludes the boundary, so that line is drawn broken
  • y ⩾ 0 gives the line y = 0, which is the x-axis itself, drawn solid
  • Every line is drawn right across the grid, not just along the edge of the region, because each one is credited separately