Video: Solving Quadratic Inequalities in One Variable

Find all solutions to the inequality 𝑥² − 7𝑥 + 12 >0. Write your answer as an interval.

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Video Transcript

Find all solutions to the inequality 𝑥 squared minus seven 𝑥 plus 12 is greater than zero. Write your answer as an interval.

First thing that we need to do is to factor the left-hand side. Two numbers that multiply to be 12 and add to be negative seven are negative three and negative four. So either 𝑥 minus three is greater than zero and 𝑥 minus four is less than zero or 𝑥 minus three is less than zero and 𝑥 minus four is greater than zero.

So we have 𝑥 is greater than three and 𝑥 is less than four or 𝑥 is less than three and 𝑥 is greater than four. You can’t be less than three and greater than four, but you can be greater than three and less than four. So looking at our graph, we know that it’s a positive leading coefficient. So it opens upwards. And our zeros are at three and four. So when is this greater than zero? Everywhere except in between three, four. So our solution will be all reals minus three, four. Because at three and at four, we’re not greater than zero. We’re actually equal to zero, so we wanna take those out.

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