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We're told to graph all possible values for h on the

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number line.

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And this is a especially interesting inequality because

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we also have an absolute value here.

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So the way we're going to do it, we're going to solve this

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inequality in terms of the absolute value of h, and from

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there we can solve it for h.

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So let's just get the absolute value of h on

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one side of the equation.

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So the easiest way to do this is to add 19 and 1/2 to both

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sides of this equation.

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I often like putting that as an improper fraction, but 1/2

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is pretty easy to deal with, so let's add 19 and 1/2 to

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both sides of this inequality.

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Did I just say equation?

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It's an inequality, not an equation, it's an inequality

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sign, not an equal sign.

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So plus 19 and 1/2.

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On the left-hand side, these guys obviously cancel out,

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that was the whole point, and we are left with the absolute

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value of h on the left-hand side is less than.

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And then if we have 19 and 1/2, essentially minus 12, 19

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minus 12 is 7, so it's going to be 7 and 1/2.

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So now we have that the absolute value of h is less

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than 7 and 1/2.

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So what does this tell us?

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This means that the distance, another way to interpret

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this-- remember, absolute value is the same thing as

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distance from 0-- so another way to interpret this

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statement is that the distance from h to 0 has to be less

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than 7 and 1/2.

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So what values of h are going to have less than a distance

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from 7 and 1/2?

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Well, it could be less than 7 and 1/2 and greater than 0, or

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equal to 0.

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So let me put it this way.

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So h could be less than 7 and 1/2.

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But if it gets too far negative, if it goes to

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negative 3, we're cool, negative 4, negative 5,

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negative 6, negative 7, we're still cool, but then at

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negative 8, all of a sudden the absolute value isn't going

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to be less than this.

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So it also has to be greater than negative 7 and 1/2.

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If you give me any number in this interval, its absolute

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value is going to be less than 7 and 1/2 because all of these

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numbers are less than 7 and 1/2 away from 0.

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Let me draw it on the number line, which they

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want us to do anyway.

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So if this is the number line right there, that is 0, and we

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draw some points, let's say that this is 7, that is 8,

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that is negative 7, that is negative 8.

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What numbers are less than 7 and 1/2 away from 0?

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Well, you have everything all the way up to-- 7 and 1/2 is

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exactly 7 and 1/2 away, so you can't count that, so 7 and

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1/2, you'll put a circle around it.

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Same thing true for negative 7 and 1/2, the absolute value,

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it's exactly 7 and 1/2 away.

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We have to be less than 7 and 1/2 away, so neither of those

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points are going to be included, positive 7 and 1/2

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or negative 7 and 1/2.

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Now, everything in between is less than 7 and 1/2 away from

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0, so everything else counts.

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Everything outside of it is clearly more than 7 and 1/2

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away from 0.

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And we're done.

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