Video: Finding the Solution Set of an Absolute Value Inequality Algebraically

Use the graph to find the solution set of the inequality ๐‘“(๐‘ฅ) < ๐‘”(๐‘ฅ) in โ„.

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Video Transcript

Use the graph to find the solution set of the inequality ๐‘“ of ๐‘ฅ is less than ๐‘” of ๐‘ฅ in the real numbers.

If this were an equation instead of an inequality, it would be where these two graphs intersect that would be our answer cause itโ€™s where theyโ€™re equal to each other. However, this is an inequality. Thatโ€™s like less than, greater than, and so on.

Visually, we can see that the ๐‘“ of ๐‘ฅ graph is less than the ๐‘” of ๐‘ฅ graph, here in the pink. Thatโ€™s where itโ€™s below that line, so between zero and six. And we must use parentheses, because at zero those lines are equal to each other. And at six, those lines are equal to each other, so we use a parenthesis.

We can always double-check our work with some algebra. So, when is the absolute value of ๐‘ฅ minus three less than three? Well, when weโ€™re working with absolute values, what could you plug into an absolute value to get three? Well, you could plug in positive three, and you could plug in negative three. So, we keep our sign, the less than sign, three. And then, we change our sign and do the negative three cause we said the three and the negative three, when you take the absolute value, would give you three, resulting in ๐‘ฅ is less than six and greater than zero, which is what our interval was, from zero to six. Numbers less than six and greater than zero are between zero and six.

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