# How do you solve and graph abs(3h-3)<12?

Jul 14, 2017

$- 5 < h < + 5$

Full explanation given

#### Explanation:

$\textcolor{b l u e}{\text{Preamble}}$

The | | are special brackets. They indicate that whatever is inside them is to be considered as positive.

To emphasize this point, suppose that instead of the < we had = giving us:

$| 3 h - 3 | = 12$

This would have to be the same as

$| \pm 12 | = + 12$

So you would get 2 values for $h$
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$\textcolor{b l u e}{\text{Answering the question}}$

Given:$\text{ } | 3 h - 3 | < 12$

Treating the | | as behaving just like brackets factor out the 3

$3 | h - 1 | < 12$

divide both sides by 3

$| h - 1 | < 4$

Just for the moment consider this as

$h - 1 < 4$

$h < 5$

Now put back the 'absolute' brackets

$| h | < 5$

Write as

$| \pm h | < 5$
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For this to work we have the condition:

$- 5 < h < + 5$

The plot of this on the number line looks like this:-

The 'hollow blobs' on the and of the red line (above) indicate that the actual values of 5 and -5 are not included.

The feasible region for $x \mathmr{and} y$ is the shaded area in the graph below.