WEBVTT



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Okay, what does this have written on the board?

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It's an absolute-value inequality.

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And it has two variables.

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So what does that tell us?

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The solutions are ordered pairs, so we graph the solution set on a coordinate plane.

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This is an equation.

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What does the graph of this equation look like?

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We graphed these last week?

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Oh, oh, oh.

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Four to the right.

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It's a V looking thing.

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Oh yeah.

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Is it a V or an upside down V?

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No, just a V.

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A V because of what?

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Because the slope is positive.

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Positive coefficient there.

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OK.

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Let's graph this for a moment.

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We're going to use some transformations.

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Where's the vertex going to be?

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The vertex is at 3, negative 4.

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In vertex form, use the opposite of the number inside the absolute value and the

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number outside.

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The boundary is a V with slopes 2 and negative 2.

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Starting at the vertex, use rise 2 and run 1 on each arm.

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The boundary graph is part of the solution set because the inequality includes

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equality.

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Are all of those ordered pairs also solutions to the inequality?

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Definitely, because what?

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Every ordered pair on the solid boundary satisfies the equality part of the

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inequality.

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Okay.

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If the inequality were strictly less than, the boundary would be dashed instead of

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solid.

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Okay, so this is just some of the solutions

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There are many more solutions beyond the boundary.

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Shade one side of the boundary to show all remaining solutions.

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And how do I decide which side to shade?

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Use a test point to decide which side of the boundary to shade.

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Let's do a test point.

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The origin, zero comma zero, is a convenient test point when it is not on the

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boundary.

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Test zero less than or equal to 2 times the absolute value of zero minus 3, minus 4.

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The absolute value is 3, so the right side is 6 minus 4, which is 2.

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Zero is less than or equal to 2, so the origin is in the solution region.

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Shade the entire side of the V that contains the origin.

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Every point in the shaded green region is a solution to the two-variable

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absolute-value inequality.

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Graph the boundary with the same transformations used for an equation, then shade

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the additional solutions.

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Okay, so what am I going to draw first?

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The boundary equation is y equals negative one half times the absolute value of x

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plus 1, plus 1.

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Use a dashed boundary because the inequality is strict and does not include

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equality.

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Points on a dashed boundary are not solutions.

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The vertex is at negative 1, 1.

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The negative coefficient reflects the graph, so the boundary is an upside-down V.

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The two slopes are one half and negative one half.

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Now choose a test point; the origin works because it is not on the boundary.

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Can we choose zero, zero?

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Sure, as long as it's not on that line.

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If it's on the line, we've got to choose any other ordered pair.

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Test whether 0 is greater than negative one half times the absolute value of 0 plus

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1, plus 1.

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The right side is one half, so 0 is not greater than one half.

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The origin is not a solution, so shade the region on the other side of the dashed V.