Math AnalysisDate: 09/13/97 at 19:38:19 From: Raymond Subject: Math Analysis In the following problems I have to figure out the boundaries for the inequality. I get as far as breaking up the inequality but from there I can't figure out what step to do next. The problems are as follows: 1) x(x-2/3)(x+1/3)<0 2) 4x^3-6x^2 is less than or equal to 0. 3) 4x(x^2-6)/x^2-4 is less than 0. I appreciate your help! Date: 09/14/97 at 08:57:03 From: Doctor Anthony Subject: Re: Math Analysis > 1) x(x-2/3)(x+1/3)<0 When the expression is factorized like this the answer is particularly easy to write down. We can see that the lefthand side is a cubic polynomial, with the term in x^3 being positive. Such a curve has a characteristic S shape with the the lefthand end disappearing to -infinity for both x and y, the righthand end to + infinity for x and y, and in the middle we get a maximum turning point first; then, as x increases, a minimum turning point before the curve bends up and goes off to +infinity. Now this curve cuts the x axis (i.e. making the whole expression = 0) at x = -1/3, at x = 0 and at x = 2/3. With this picture in mind, it is clear that the curve is negative if x < -1/3, also if x lies between 0 and 2/3; otherwise it is positive. So the inequality is satisfied when: x < -1/3 and 0 < x < 2/3 > 2) 4x^3-6x^2 is less than or equal to 0. Again take factors 2x^2(2x-3) <= 0 This is the same shape as that described before for a cubic polynomial, except that the maximum turning point at x = 0 just touches the x axis and does not go above it before bending down towards the minimum turning point. After the minimum point the curve bends up and crosses the x axis at x = 3/2. So again we can picture where the curve lies below the x axis, thereby satisfying the condition that it be less than 0. Required condition is x <= 3/2 > 3) 4x(x^2-6)/x^2-4 is less than 0. Although one of the terms in this expression divides the other two, we get the same result if we think of it as multiplying the other two. This is because you get a change of sign whether you multiply or divide by a negative quantity, and it is easy to visualize a curve's shape if all the terms multiply. So I am changing this to a different problem with all terms multiplying, but the answers will be the same. 4x(x^2-6)(x^2-4) <= 0 carry out further factorizing as possible: 4x(x-sqrt^6))(x+sqrt(6))(x-2)(x+2) <= 0 Now the places where the curve cuts the x axis in ascending order are: -sqrt(6), -2, 0, +2, +sqrt(6) Now you must imagine the 5th power curve behaving like the cubic already described, but with an extra hump. So coming from the left, the curve cuts the x axis from below at -sqrt(6). Thus the inequality is satisfied with x <= -sqrt(6). We now get a positive region between -sqrt(6) and -2. Then a negative region between -2 and 0. So the inequality is satisfied with -2 <= x < = 0. From here the curve goes positive again between 0 and +2, but from +2 to +sqrt(6) the curve is again below the x axis. So the inequality is satisfied with 2 <= x <= sqrt(6). Putting all these facts together we get the inequality satisfied when: x <= -sqrt(6), -2 <= x <= 0, 2 <= x <= sqrt(6) -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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