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Thursday, July 16, 2020

Linear Inequalities in Two Variables - Notes

Here are the notes for the Linear Inequalities in Two Variables Section of the Linear Programming Unit.  Let me know if you have any questions or need further assistance.

Graph inequalities in the Cartesian coordinate system, i.e. ax + by greater than or equal to c or y < mx + b. Steps: Graph the line y = mx + b or ax + by = c. Find the x-intercept and y-intercept. Draw the line between these points. Pick a point not on the line. Substitute this point into the inequality equation. If the result is true, shade where the point came from. If the result is false, do not shade where the point came from. Notation: If the inequality is greater than or equal to or less than or equal to use a solid line. If the inequality is > or < use a dashed line. Example: Graph y greater than or equal to 1. Draw the Cartesian coordinate system. Draw a solid line at y = 1. Then, shade above the line y = 1. Since 2 is greater than or equal to 1, the point (0,2) is true and we shade there. Since 0 is not greater than or equal to 1, the point (0, 0) is not included. We do not shade there.  

Note: For the last sentence in the previous image, the point should be (0, 0), not (0, 1).  My apologies for any confusion.

Graph x + 2y > 4. Step 1: Consider x + 2y = 4. x-intercept: Put y = 0 into x + 2y = 4 to get x = 4. So, (4, 0) is on the line. y-intercept: Put x = 0 into x + 2y + 4 to get 2y = 4, i.e. y = 2. So, (0, 2) is on the line. Step 2: Graph x + 2y = 4 (with a dashed line due to >) Step 3: Check points above and below the line x + 2y = 4. Check (0,0): Put (0,0) into x + 2y > 4. Get 0 + 2(0) = 0 > 4, which is false. Don't shade where (0,0) is at. Check (0,5): Put (0, 5) into x + 2y > 4. Get 0 + 2(5) = 10 > 4, which is true. Shade where (0, 5) is at. If you have time, graph -x + 2y < 2.

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