Integrated Math 2–3 Honors — Graphing Absolute-Value Inequalities in Two Variables
Corrected lesson transcript


## 0:00 — Graphing Absolute-Value Inequalities


## 0:05 — What the Solution Set Represents

[0:05] Okay, what does this have written on the board?
[0:13] It's an absolute-value inequality.
[0:17] And it has two variables.
[0:19] So what does that tell us?
[0:20] The solutions are ordered pairs, so we graph the solution set on a coordinate plane.

## 0:33 — Example 1: Graph a Solid V Boundary

[0:33] This is an equation.
[0:37] What does the graph of this equation look like?
[0:44] We graphed these last week?
[0:45] Oh, oh, oh.
[0:46] Four to the right.
[0:48] It's a V looking thing.
[0:49] Oh yeah.
[0:50] Is it a V or an upside down V?
[0:52] No, just a V.
[0:54] A V because of what?
[0:55] Because the slope is positive.
[0:57] Positive coefficient there.
[1:00] OK.
[1:03] Let's graph this for a moment.
[1:06] We're going to use some transformations.
[1:08] Where's the vertex going to be?
[1:13] The vertex is at 3, negative 4.
[1:15] In vertex form, use the opposite of the number inside the absolute value and the number outside.
[1:22] The boundary is a V with slopes 2 and negative 2.
[1:29] Starting at the vertex, use rise 2 and run 1 on each arm.
[1:37] The boundary graph is part of the solution set because the inequality includes equality.
[1:44] Are all of those ordered pairs also solutions to the inequality?
[1:51] Definitely, because what?
[1:52] Every ordered pair on the solid boundary satisfies the equality part of the inequality.
[1:57] Okay.
[1:58] If the inequality were strictly less than, the boundary would be dashed instead of solid.
[2:08] Okay, so this is just some of the solutions
[2:13] There are many more solutions beyond the boundary.
[2:17] Shade one side of the boundary to show all remaining solutions.
[2:27] And how do I decide which side to shade?

## 2:30 — Use a Test Point and Shade

[2:30] Use a test point to decide which side of the boundary to shade.
[2:31] Let's do a test point.
[2:35] The origin, zero comma zero, is a convenient test point when it is not on the boundary.
[2:40] Test zero less than or equal to 2 times the absolute value of zero minus 3, minus 4.
[2:46] The absolute value is 3, so the right side is 6 minus 4, which is 2.
[2:50] Zero is less than or equal to 2, so the origin is in the solution region.
[2:57] Shade the entire side of the V that contains the origin.
[3:05] Every point in the shaded green region is a solution to the two-variable absolute-value inequality.
[3:14] Graph the boundary with the same transformations used for an equation, then shade the additional solutions.

## 3:31 — Example 2: Graph a Dashed V Boundary

[3:31] Okay, so what am I going to draw first?
[3:38] The boundary equation is y equals negative one half times the absolute value of x plus 1, plus 1.
[3:48] Use a dashed boundary because the inequality is strict and does not include equality.
[3:55] Points on a dashed boundary are not solutions.
[4:01] The vertex is at negative 1, 1.
[4:12] The negative coefficient reflects the graph, so the boundary is an upside-down V.
[4:28] The two slopes are one half and negative one half.

## 4:34 — Test the Second Region

[4:34] Now choose a test point; the origin works because it is not on the boundary.
[4:38] Can we choose zero, zero?
[4:40] Sure, as long as it's not on that line.
[4:42] If it's on the line, we've got to choose any other ordered pair.
[4:50] Test whether 0 is greater than negative one half times the absolute value of 0 plus 1, plus 1.
[5:07] The right side is one half, so 0 is not greater than one half.
[5:13] The origin is not a solution, so shade the region on the other side of the dashed V.