0580

Solving and Graphing Inequalities

Coordinate Geometry and Graphs

Working backwards

  • The reverse question shows a region already marked R and asks which inequalities define it
  • Start by identifying the equation of every line that bounds the region, leaving the inequality signs aside for the moment
    • A vertical line is x = k and a horizontal line is y = k
    • A sloping line needs its gradient and where it meets the vertical axis, giving y = mx + c
  • Getting those equations right earns marks by itself, even before any inequality sign is decided
  • Then turn each equation into an inequality by looking at where R lies relative to that line
A shaded triangular region bounded by the solid vertical line x = 1, the solid downhill line y = −x + 7 and the broken line y = x, with a test point at (2, 4) inside it for checking each inequality sign
Source: Interpreting graphical inequalities by Save My Exams
  • Solid means ⩽ or ⩾ and broken means < or >, so the style of the line settles which pair of signs to use
  • Confirm every one by substituting the coordinates of a point inside R, which takes seconds and catches a reversed sign
Exam tip

Shade the side you do not want — the opposite of what feels natural — and match the line style to the sign: solid for ⩽ and ⩾, broken for < and >. Then write R in the clear patch: one mark scheme states that once R is labelled, the lines around it are taken as your boundaries and the shading is ignored, so the label is worth more than neat shading.

Worked example

Finding the inequalities that define a region

A region R is bounded by a solid vertical line at x = 2, a solid horizontal line at y = 4, and a broken line through (0, −1) and (3, 2). R lies to the right of the vertical line, below the horizontal one, and above the sloping one. Write down the three inequalities.

Solution:

  • The vertical line gives x = 2, and R is to the right of a solid line, so x ⩾ 2
  • The horizontal line gives y = 4, and R is below a solid line, so y ⩽ 4
  • The sloping line rises 3 for a run of 3, so its gradient is 1, and it meets the vertical axis at −1, giving y = x − 1
  • R is above that line and the line is broken, so y > x − 1
  • The three inequalities are x ⩾ 2, y ⩽ 4 and y > x − 1
  • Check with (3, 3), a point inside R: 3 ⩾ 2, 3 ⩽ 4 and 3 > 2 all hold