Integrated Math 2–3 Honors — Solving Absolute-Value Inequalities
Corrected lesson transcript


## 0:00 — Solving Absolute-Value Inequalities


## 0:05 — What an Absolute-Value Inequality Represents

[0:05] Okay, what is this thing I've written on the board called? Absolute-value inequality. Absolute-value inequality. Yeah.
[0:23] Is it true, false, or open?
[0:26] Open.
[0:28] Are the solutions numbers or pairs?
[0:33] Numbers because?
[0:35] There's one variable.
[0:36] There's only one variable.
[0:36] How did we solve this equation before?
[0:43] We split it up into its two cases.
[0:46] What are the two cases?
[0:50] And it's because this thing is a piecewise function, right?
[0:54] Yes.
[0:54] It is either x or negative x, depending on whether x is greater than or equal to zero or x is less than zero.
[1:01] These inequalities are gonna have a lot more than a couple
[1:06] Okay, so let me show you

## 1:09 — Why Less Than Becomes an AND Inequality

[1:09] Now we're going to do it, then there's going to be kind of a shortcut.
[1:12] But here's the whole idea.
[1:15] First, the absolute value of x equals x when x is greater than or equal to zero, and negative x when x is less than zero.
[1:27] That is the piecewise definition of absolute value.
[1:34] Okay, so we're going to split this up into two cases.
[1:39] If x is greater than or equal to zero.
[1:43] So since x is greater or equal to zero, what can I replace the absolute value of x with?
[1:52] Just this.
[1:55] So I go x is less than or equal to 5.
[2:03] Then the inequality gives x less than or equal to 5.
[2:08] However, it has to be both greater than or equal to 0 AND less than or equal to 5.
[2:18] So it's a compound inequality basically.
[2:20] What should all be these numbers?
[2:23] 0 to 5.
[2:25] So it's all these numbers.
[2:30] So that's part of our solution set.
[2:32] The other part is if x is less than zero.
[2:39] If x is less than zero, what do I replace this with?
[2:44] It's equal to negative x.
[2:54] And then I gotta get rid of that negative and how do I do that?
[2:58] Multiply the negative one.
[3:00] Multiply it by negative one and when we multiply it by a negative we have to flip.
[3:06] This is supposed to be negative 5, so the numbers are less than zero and greater than or equal to negative 5.
[3:11] negative 5 so all the numbers are less than 0 and greater than negative 5 is
[3:21] this solution said
[3:23] I'll draw those numbers.
[3:27] So the inequality solution set is this and this combined.
[3:38] So the two pieces are the interval from negative 5 to 0 union the interval from 0 to 5.
[4:00] Is there a reason why I just write negative 5 to 5?
[4:04] We're about to get to that, yes.
[4:07] Combining those two pieces gives the single interval from negative 5 to 5.
[4:15] Agree everything in red and green is all of these numbers this set
[4:20] The equivalent compound inequality is negative 5 less than or equal to x less than or equal to 5.
[4:26] The solution set to this is this exactly right here.
[4:33] Okay, so the shortcut, whenever we're given any absolute value is less than or less than
[4:42] or equal to a number, we're not going to go through both of these steps because the shortcut
[4:49] Rewrite the expression without absolute-value bars, place the same inequality on the left, and use the opposite endpoint.
[5:00] For this example, the result is negative 5 less than or equal to x less than or equal to 5.
[5:08] This is an AND compound inequality: x must be less than or equal to 5 and greater than or equal to negative 5.
[5:17] equal to 5 and greater than or equal to negative 5.

## 5:22 — Example: Solve an Inward-Pointing Inequality

[5:22] So let's try another one real quick.
[5:27] It will make it a little bit complicated.
[5:38] And here we don't have just X, but it doesn't really matter.
[5:44] The shortcut makes a two-for-one AND compound inequality.
[5:49] So we have 3x plus 4 less than or equal to 10.
[5:56] Put the same less-than-or-equal-to symbol on the left side.
[6:01] And we put the opposite number over here.
[6:07] Now we got a compound and inequality.
[6:12] We solve it just like we did we learn that on Monday something like that.
[6:16] Okay so we subtract four now.
[6:19] The solution is negative fourteen-thirds less than or equal to x less than or equal to 2.
[6:29] If we wanted to graph it, we could.
[6:39] Fourteen-thirds is four and two-thirds.
[6:42] Four and two-thirds.
[6:43] Four and two-thirds.
[6:43] That endpoint is negative four and two-thirds.
[7:04] Use the AND shortcut when the inequality symbols point inward toward the expression.
[7:15] When the inequality points away, use the OR shortcut instead.
[7:21] Points away, we're about to do in a second.

## 7:23 — Why Greater Than Becomes an OR Inequality

[7:23] So let me show you the other way.
[7:27] Or the other thing.
[7:29] Let's keep it simple at first.
[7:31] Okay, again we're going to split this up into two cases.
[7:36] If x is greater than or equal to zero.
[7:40] If x is greater or equal to zero, the absolute value of x is simply x.
[7:47] The conditions reduce to x greater than or equal to 4.
[7:54] What numbers are both greater than or equal to zero and greater than or equal to 4?
[8:02] Oh, numbers are greater or equal to 4.
[8:04] Yeah, just the numbers greater or equal to 4.
[8:09] Like the number 2, this is true but this is not true.
[8:13] So it has to be both this and this, which is just basically the solutions after this.
[8:19] Then if x is less than 0.
[8:31] Since x is less than zero, what is the absolute value of x?
[8:37] It's negative x.
[8:43] And we divide by negative one, divide by negative one, and flip.
[8:50] Here's the negative 4.
[8:55] The conditions reduce to x less than or equal to negative 4.
[9:04] All the ones that fulfill that.
[9:06] That interval runs from negative infinity to negative 4, including negative 4.
[9:14] The complete solution is the union of the two rays.
[9:28] Write two separate intervals and combine them with the union symbol.
[9:31] We can put the union sign to combine into one set.
[9:34] If you wanted to graph them.
[9:40] It intuitively should make sense.
[9:46] Right back in elementary school they said the absolute value is the distance from the
[9:50] origin.
[9:52] Every solution is at least 4 units from the origin.
[9:59] The distance of each number in either ray is greater than or equal to 4.
[10:05] Okay, so we end up with two different sets like that.
[10:12] Where this, right here at x, this wasn't a good one, this one, simple.
[10:19] The absolute value less than or equal to 5.
[10:23] So the distance from the origin is less than or equal to 5, which are all these numbers.
[10:29] Okay, we don't care if it's the left or the right.
[10:30] When the inequality points away from the absolute-value expression, use the OR shortcut.
[10:46] from the absolute, sorry, the inequality is pointing away from the absolute value.
[10:51] You rewrite the same thing just like we did for the other one.
[10:55] Write x greater than or equal to 4, or x less than or equal to negative 4.
[11:08] 4.
[11:11] And then this is a compound OR inequality in any solid in a second.

## 11:16 — Example: Solve an Outward-Pointing Inequality

[11:16] So let's do another example.
[11:22] It's a little bit more complicated.
[11:24] This is the two-part OR form because the inequality points away from the absolute-value expression.
[11:29] Write two separate inequalities.
[11:31] The inequality points away, so use OR.
[11:34] Right distance is greater. It's kind of two separate things
[11:38] Copy the expression inside the bars: 2x plus 3 is greater than 5.
[11:45] And we write or
[11:47] For the other branch, copy 2x plus 3.
[11:51] Then reverse the inequality and negate the endpoint.
[11:54] positive sign
[11:56] The second branch is 2x plus 3 less than negative 5.
[12:01] These we have to solve separately. We can't do it for one like the previous example.
[12:09] The boundary values are negative 4 and 1.
[12:23] In interval notation, the solution is negative infinity to negative 4, union 1 to infinity.
[12:31] OK, so again, the two shortcuts.
[12:37] If the inequality points away, write two separate inequalities joined by OR.
[12:41] Write it without the absolute value.
[12:43] Copy the expression inside the absolute-value bars.
[12:45] Then reverse the inequality and negate the endpoint for the second branch.
[12:47] The sign of the inequality sign and the positive to negative.
[12:52] And solve them separately.

## 12:53 — Multi-Step AND Example

[12:53] Let's try another one.
[12:55] Okay.
[13:03] First subtract 5 from both sides.
[13:07] Subtract 5.
[13:08] Subtract 5.
[13:08] This is the two-for-one AND form.
[13:13] The AND.
[13:16] Write negative 2 less than or equal to 6 minus 2x less than or equal to 2.
[13:26] And then we put the same sign on the left side and we flip.
[13:33] Okay, so that's over there.
[13:37] Subtract 6 from all three parts.
[13:44] We're going to be real careful here.
[14:00] When we divide by the negative 2, what do we have to do?
[14:03] Flip and flip.
[14:04] Okay.
[14:08] Reverse both inequalities, then reorder the chain so the smaller endpoint is on the left.
[14:14] both inequalities. So I'm going to put the 2 over here.
[14:19] The result is 2 less than or equal to x less than or equal to 4.
[14:22] and a point to the x, and a 4.
[14:31] So the solution is every number between 2 and 4, including both endpoints.

## 14:34 — Negative-Coefficient OR Example

[14:34] Use negative 12 on the right side.
[14:38] That's what I wanted.
[14:45] What do we have to do first?
[14:51] Very important.
[14:52] Isolate the absolute-value expression before applying either shortcut.
[14:56] Divide by negative 3.
[15:04] Dividing by negative 3 reverses the inequality, giving absolute value of x minus 5 greater than or equal to 4.
[15:10] We've got to flip it around.
[15:18] And is this the AND or the OR compound?
[15:22] OR.
[15:23] This is an OR inequality, so write two separate branches.
[15:26] The branches are x minus 5 greater than or equal to 4, or x minus 5 less than or equal to negative 4.
[15:39] Okay.
[15:50] Okay.
[15:51] The solutions are x greater than or equal to 9, or x less than or equal to 1.
[16:05] In interval notation: negative infinity to 1, union 9 to infinity.