Graphing a Linear Inequality: Boundary Line and Shaded Side

0:00How do we graph 2x plus 3y is less than or equal to 12?

0:06Its solutions are all the points that make it true: a whole region.

0:11Step 1: the boundary line, where both sides are equal.

0:16Intercepts are where the line crosses the axes. Set y to 0: 2x is 12, so x is 6. Set x to 0: 3y is 12, so y is 4.

0:31It runs 6 and drops 4, so its slope, rise over run, is negative 2 thirds.

0:38The or equal to part means points on the line count, so draw it solid.

0:44Step 2: test a point off the line, like 0, 0: 2 times 0 plus 3 times 0 is 0.

0:530 is less than or equal to 12, true, so shade the origin's side.

0:59Now add y is greater than x minus 2. Its boundary has y-intercept negative 2 and slope 1.

1:08Greater than leaves out the line itself, so draw it dashed.

1:13Test 0, 0: 0 is greater than 0 minus 2, true, so shade the origin's side.

1:21Together they form a system: both are true only where the shadings overlap.

1:27The corner is where the lines cross. Substitute x minus 2 for y in the first equation.

1:34Then 5x equals 18, so x is 3.6 and y is 1.6.

1:42Check in the first equation: 7.2 plus 4.8 is 12.

1:48It sits on the dashed line, so it is not a solution: an open dot.

1:54A common mistake is shading the wrong side. Always test a point, even with software.

2:00To recap: draw the boundary, solid or dashed, test a point, and shade the true side.