Theory question 107
InequalitiesIllustrate the following on graph paper and shade the region which satisfies all the three inequalities at the same time.
−x + 5y ≤ 10
3x − 4y ≤ 8
and x > −1
Show the model answer
Draw the boundary lines: −x + 5y = 10 (through (−10, 0) and (0, 2)); 3x − 4y = 8 (through (0, −2) and (8/3, 0)); x = −1 (broken line, since the inequality is strict).
−x + 5y ≤ 10: region below the first line (the origin satisfies it).
3x − 4y ≤ 8: region above the second line (the origin satisfies it).
x > −1: region to the right of x = −1.
The required region is the triangle with vertices (−1, 1.8), (−1, −2.75) and about (8.3, 3.5) where the two slanting lines meet; shade it (or shade the unwanted regions, and say which convention is used).
Test the origin in each inequality to decide which side to keep.