Video: Solving Quadratic Inequalities in One Variable at a Specific Interval Graphically

Using the graph, find the values of 𝑥 for which 𝑓(𝑥) ⩾ 0.

02:31

Video Transcript

Using the graph, find the values of 𝑥 for which 𝑓 of 𝑥 is greater than or equal to zero. And we’re given that 𝑓 of 𝑥 is equal to minus 𝑥 squared plus five 𝑥. And we’re also given the graph which shows 𝑦 equals 𝑓 of 𝑥, which actually maps out that quadratic. So this is a symmetrical parabola, cuts the 𝑥-axis at zero and five, and also cuts the 𝑦-axis at zero. So we’ve gotta find out when 𝑓 of 𝑥 is greater than zero; in other words, when the 𝑦-coordinate is greater than or equal to zero.

So with 𝑦 equals 𝑓 of 𝑥, and we’re looking for when 𝑓 of 𝑥 is greater than or equal to zero, we’re really- we’re looking for are all the points on that curve where the 𝑦-coordinate is greater than or equal to zero. Well here at zero and here at five on the 𝑥-axis, the 𝑦-coordinate is zero. And on all these points on the curve in between, the 𝑦-coordinate is greater than zero. So that’s the region that we’re interested in. So we wanna know the 𝑥-coordinates that generate those. Well zero, five, everything in between in terms of the 𝑥-coordinate, so from zero to five, and everything outside that region. So bigger than five onwards or less than zero onwards down to negative infinity, that’s not included in our region because these parts of the curve are not greater than or equal to zero. So we could write that like this, zero is less than or equal to 𝑥 is less than or equal to five.

But we could also write it in interval format. So the critical values are zero and five. That’s either end of the interval. Now zero is included, so we need to clu- include a square bracket at the end. And five is included, so we put the square brackets around that end. So that’s in interval format. So we could also put it in set notation. So we’ve got the set of 𝑥 such that 𝑥 is a real number between zero and five. So the process then was to have a look at the graph and identify all the points on the graph which match the criteria that we’re looking for. In this case, it was 𝑓 of 𝑥 is greater than or equal to zero. So it’s then a question of finding which 𝑥-coordinates match the criteria and which 𝑥-coordinates don’t match that criteria, and then just kind of summarising that in one of these formats: the appropriate format whether it’s the inequality format, the interval format, or the set format depending on what the question asks for.

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