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Okay, what is this thing I've written on the board called? Absolute-value

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inequality. Absolute-value inequality. Yeah.

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Is it true, false, or open?

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Open.

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Are the solutions numbers or pairs?

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Numbers because?

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There's one variable.

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There's only one variable.

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How did we solve this equation before?

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We split it up into its two cases.

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What are the two cases?

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And it's because this thing is a piecewise function, right?

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Yes.

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It is either x or negative x, depending on whether x is greater than or equal to

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zero or x is less than zero.

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These inequalities are gonna have a lot more than a couple

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Okay, so let me show you

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Now we're going to do it, then there's going to be kind of a shortcut.

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But here's the whole idea.

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First, the absolute value of x equals x when x is greater than or equal to zero, and

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negative x when x is less than zero.

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That is the piecewise definition of absolute value.

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Okay, so we're going to split this up into two cases.

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If x is greater than or equal to zero.

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So since x is greater or equal to zero, what can I replace the absolute value of x

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with?

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Just this.

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So I go x is less than or equal to 5.

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Then the inequality gives x less than or equal to 5.

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However, it has to be both greater than or equal to 0 AND less than or equal to 5.

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So it's a compound inequality basically.

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What should all be these numbers?

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0 to 5.

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So it's all these numbers.

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So that's part of our solution set.

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The other part is if x is less than zero.

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If x is less than zero, what do I replace this with?

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It's equal to negative x.

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And then I gotta get rid of that negative and how do I do that?

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Multiply the negative one.

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Multiply it by negative one and when we multiply it by a negative we have to flip.

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This is supposed to be negative 5, so the numbers are less than zero and greater

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than or equal to negative 5.

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negative 5 so all the numbers are less than 0 and greater than negative 5 is

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this solution said

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I'll draw those numbers.

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So the inequality solution set is this and this combined.

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So the two pieces are the interval from negative 5 to 0 union the interval from 0 to

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5.

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Is there a reason why I just write negative 5 to 5?

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We're about to get to that, yes.

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Combining those two pieces gives the single interval from negative 5 to 5.

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Agree everything in red and green is all of these numbers this set

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The equivalent compound inequality is negative 5 less than or equal to x less than

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or equal to 5.

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The solution set to this is this exactly right here.

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Okay, so the shortcut, whenever we're given any absolute value is less than or less

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than

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or equal to a number, we're not going to go through both of these steps because the

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shortcut

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Rewrite the expression without absolute-value bars, place the same inequality on the

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left, and use the opposite endpoint.

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For this example, the result is negative 5 less than or equal to x less than or

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equal to 5.

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This is an AND compound inequality: x must be less than or equal to 5 and greater

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than or equal to negative 5.

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equal to 5 and greater than or equal to negative 5.

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So let's try another one real quick.

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It will make it a little bit complicated.

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And here we don't have just X, but it doesn't really matter.

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The shortcut makes a two-for-one AND compound inequality.

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So we have 3x plus 4 less than or equal to 10.

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Put the same less-than-or-equal-to symbol on the left side.

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And we put the opposite number over here.

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Now we got a compound and inequality.

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We solve it just like we did we learn that on Monday something like that.

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Okay so we subtract four now.

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The solution is negative fourteen-thirds less than or equal to x less than or equal

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to 2.

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If we wanted to graph it, we could.

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Fourteen-thirds is four and two-thirds.

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Four and two-thirds.

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Four and two-thirds.

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That endpoint is negative four and two-thirds.

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Use the AND shortcut when the inequality symbols point inward toward the expression.

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When the inequality points away, use the OR shortcut instead.

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Points away, we're about to do in a second.

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So let me show you the other way.

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Or the other thing.

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Let's keep it simple at first.

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Okay, again we're going to split this up into two cases.

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If x is greater than or equal to zero.

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If x is greater or equal to zero, the absolute value of x is simply x.

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The conditions reduce to x greater than or equal to 4.

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What numbers are both greater than or equal to zero and greater than or equal to 4?

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Oh, numbers are greater or equal to 4.

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Yeah, just the numbers greater or equal to 4.

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Like the number 2, this is true but this is not true.

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So it has to be both this and this, which is just basically the solutions after

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this.

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Then if x is less than 0.

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Since x is less than zero, what is the absolute value of x?

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It's negative x.

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And we divide by negative one, divide by negative one, and flip.

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Here's the negative 4.

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The conditions reduce to x less than or equal to negative 4.

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All the ones that fulfill that.

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That interval runs from negative infinity to negative 4, including negative 4.

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The complete solution is the union of the two rays.

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Write two separate intervals and combine them with the union symbol.

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We can put the union sign to combine into one set.

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If you wanted to graph them.

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It intuitively should make sense.

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Right back in elementary school they said the absolute value is the distance from

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the

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origin.

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Every solution is at least 4 units from the origin.

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The distance of each number in either ray is greater than or equal to 4.

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Okay, so we end up with two different sets like that.

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Where this, right here at x, this wasn't a good one, this one, simple.

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The absolute value less than or equal to 5.

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So the distance from the origin is less than or equal to 5, which are all these

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numbers.

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Okay, we don't care if it's the left or the right.

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When the inequality points away from the absolute-value expression, use the OR

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shortcut.

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from the absolute, sorry, the inequality is pointing away from the absolute value.

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You rewrite the same thing just like we did for the other one.

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Write x greater than or equal to 4, or x less than or equal to negative 4.

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4.

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And then this is a compound OR inequality in any solid in a second.

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So let's do another example.

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It's a little bit more complicated.

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This is the two-part OR form because the inequality points away from the

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absolute-value expression.

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Write two separate inequalities.

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The inequality points away, so use OR.

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Right distance is greater. It's kind of two separate things

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Copy the expression inside the bars: 2x plus 3 is greater than 5.

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And we write or

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For the other branch, copy 2x plus 3.

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Then reverse the inequality and negate the endpoint.

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positive sign

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The second branch is 2x plus 3 less than negative 5.

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These we have to solve separately. We can't do it for one like the previous example.

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The boundary values are negative 4 and 1.

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In interval notation, the solution is negative infinity to negative 4, union 1 to

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infinity.

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OK, so again, the two shortcuts.

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If the inequality points away, write two separate inequalities joined by OR.

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Write it without the absolute value.

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Copy the expression inside the absolute-value bars.

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Then reverse the inequality and negate the endpoint for the second branch.

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The sign of the inequality sign and the positive to negative.

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And solve them separately.

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Let's try another one.

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Okay.

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First subtract 5 from both sides.

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Subtract 5.

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Subtract 5.

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This is the two-for-one AND form.

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The AND.

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Write negative 2 less than or equal to 6 minus 2x less than or equal to 2.

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And then we put the same sign on the left side and we flip.

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Okay, so that's over there.

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Subtract 6 from all three parts.

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We're going to be real careful here.

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When we divide by the negative 2, what do we have to do?

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Flip and flip.

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Okay.

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Reverse both inequalities, then reorder the chain so the smaller endpoint is on the

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left.

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both inequalities. So I'm going to put the 2 over here.

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The result is 2 less than or equal to x less than or equal to 4.

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and a point to the x, and a 4.

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So the solution is every number between 2 and 4, including both endpoints.

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Use negative 12 on the right side.

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That's what I wanted.

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What do we have to do first?

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Very important.

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Isolate the absolute-value expression before applying either shortcut.

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Divide by negative 3.

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Dividing by negative 3 reverses the inequality, giving absolute value of x minus 5

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greater than or equal to 4.

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We've got to flip it around.

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And is this the AND or the OR compound?

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OR.

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This is an OR inequality, so write two separate branches.

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The branches are x minus 5 greater than or equal to 4, or x minus 5 less than or

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equal to negative 4.

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Okay.

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Okay.

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The solutions are x greater than or equal to 9, or x less than or equal to 1.

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In interval notation: negative infinity to 1, union 9 to infinity.