Problem

Solve each inequality, and graph the solution set…

07:33

Problem 28 Medium Difficulty

Solve each inequality, and graph the solution set. See Example 4.
$$
(2 x+1)(3 x-2)(4 x+7)<0
$$

Answer

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Precalculus

Algebra

Beginning and Intermediate Algebra

Chapter 11

Quadratic Equations, Inequalities, and Functions

Section 8

Polynomial and Rational Inequalities

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Video Transcript

All right, we have two x plus one times three x menace to times four X plus seven and the product of those is less than zero. We want to find the solution set for this inequality. To do that, we first have to take each of these factors. Then let them equals zero so that we confined our roots. So if two X plus one is equal to zero, I will next subtract one from each side. So two X will equal negative one and we consol by dividing by two. Then we get X equals negative 1/2. That would be one of our zeros. Next, we'll take the three X minus two unleaded equal zero. I had to double sides. You get three. X equals two. Divide by three and you get X equals 2/3. And then I take the four X plus seven. Let it equal zero. Subtract seven from both sides. You have four X equals negative seven by both sides before and we get X equals negative. Seven fourths some. This is what we have to work with. Now. This isn't what we're looking for. The right. We want to know when we were less than zero, not equal a zero. So if we were to think in terms of our number line, then if I have negative too negative, 7/4 would be negative one and 3/4 and then will say we have negative one negative 1/2 zero 2/3 and then let's say one, then what's gonna be going on here? Is that negative? 7/4 negative, 1/2 and 2/3. Those are the places where our intervals air changing these air. The endpoints of our intervals are solution. Sets are going to be on either side of these values, so we have to pick some test points out. Right? So, in other words, I have to see what's going on to the left of negative 7/4 I have to see what's going on between negative seven forts and negative 1/2 between negative 1/2 and 2/3 and what's going on to the right of 2/3. So I need four test points and I think these are the most convenient ones to use, right, negative to negative 10 on one. We really could plug in any number we want, but we want toe keep it convenient to the best of our ability. So I'm gonna test X equals negative to. And when I plug negative 21 for X, that will give us two times negative too. Plus one times, three times negative too. Minus two times, four times negative too. Plus seven. And all of that's less than zero. So if this intervals part of our solution set, then this will check. So this is gonna be negative. Four plus one. That'll be negative. Three. This is gonna be negative. Six minutes to that's gonna be negative. Eight. And that's going to negative eight plus seven. That would be negative one. Well, this is going to be negative. 3 24 24 times Negative one is gonna be negative. 24 which actually is less than zero. So this indicates to me that the values to the left of negative seven force will be part of our solution set. So next I'm gonna check this interval here between negative 7/4 of negative 1/2. So minute check X equals negative one. Next. See if this works. This will give me too times negative. One plus one times three times negative one minus two times, four times negative, one plus seven. And I want to see if that's less than zero. Negative. Two plus one will be negative. One negative three minus two will be negative. Five and negative. Four plus seven will be three. Well, one times negative five is five times three is 15. 15 is definitely not less than zero. So that interval is not going to be part of our solution set. So next we're gonna test X equals zero and see what happens. This will give us to 10 0 plus one times three times zero minus two times, four times zero for seven. Well, this is gonna be one terms negative too. Times seven, which is gonna be negative 14 which is less than zero. So that indicates to me that these values between negative 1/2 positive 2/3 we will be part of our solution set. So now we have to see what's going on on this last interval here till the rate of 2/3 and I'm gonna check X equals one. And when I plug one in for X, I get two times one plus one times three times one minus two times four times one my plus seven. Well, this is going to be three times one times 11. That's going to be positive value. It's gonna be 33 which is definitely not less than zero. So that interval is not part of our solution set. So our solution sat. Looking at this number line will be the interval from negative infinity to negative seven Ford's or the interval from negative 1/2 to 2/3. That will be our solution set right there.

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