Problem

In Exercises 1–3, the graph of a quadratic functi…

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Problem 2 Medium Difficulty

In Exercises 1–3, the graph of a quadratic function ƒ is given. Use the graph to find the solution set of each equation or inequality. See Example 1.
(a) $3 x^{2}+10 x-8=0$
(b) $3 x^{2}+10 x-8 \geq 0$
(c) $3 x^{2}+10 x-8<0$

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. So we want to know when three X squared plus 10 X minus eight is equal to zero. If we were to take a look at the graph of this, it becomes pretty easy to spot. If we just sketched this out, the crafts a little bit off. But what we're gonna find out is that we have roots negative for and a positive 2/3. So those two points are the zeros there, the solution. So we would say ex equals negative four and acts equals 2/3. Now, if we take the same function three x squared plus 10 X, many say is not equal to zero, that's the one we just dead is greater than or equal to zero. Then we're looking at similar graph which have something that's doing something kind like this. Maybe a little bit more like that. But zero's air still the same. We should be at negative four and at 2/3. The difference, though here is we want to know when were greater than or equal to zero. Well, we know we're equal to zero at those values, right? We're gonna be greater than zero. When were above the access. So all of the values where we're above the access would be all of the values to the left of negative four and all of the values to the right of 2/3. Now, we also want to know where we are equal to zero. So our solution set has to include the roots. Okay, We do not want those to be open. So we're gonna be looking at the interval from negative infinity to negative four. Except we want to break it there on the four on the negative four. The bracketing indicates that that end point is included in the solution side or the interval from 2/3 to infinity. Right? This is equivalent to saying X is less than or equal to negative four or X is greater than or equal to 2/3. That's what that interval notation means in the last one. We want to consider same function. We have three X squared plus 10 x minus eight is less than zero. So notice now this says less than zero. We're not including the zeros. As part of this solution. Set them. We only want to know when we are less than zero. So again we take our function. We still have roots at X equals negative four and X equals 2/3. We are below the access on the interval when we're between those two values right between negative four and 2/3. So in other words, the X values that are between negative foreign 2/3. So using interval notation, that would be the interval from negative four, 2 2/3 and again I'm using parentheses, not brackets, because we're not including those zeros as part of our set. So now we have all three.

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