Question 478755: Can someone please help solve this equation
6x(x-2)(3x+5)<0
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! easy way to solve it is to graph it.
that's kind of like cheating, but here goes.
by looking at the graph you can see the regions that it's positive in and the regions that it's negative in.
if you want to do it by formula, then you would want to find the 0 points of equation.
set the equation equal to 0 to get:
6x(x-2)(3x+5)=0
this equation will be 0 if:
6x = 0
or:
x-2 = 0
or:
3x+5 = 0
solve for x in each of those equations and you get the overall equation being when:
x = 0
x = 2
x = -5/3
look at the graph and you'll see that's where the graph crosses the x-axis.
if you didn't have a graph to look at, you would simply know that the equation is equal to 0 when x is the values shown.
the equation you are using is a continuous function.
this means that there are no breaks in the line of the equation.
it is smooth and continuous from end to end.
this means that:
before or after or in between the 0 points of the equation, if it's negative it will stay negative, and if it's positive it will stay positive.
since you know the 0 points, just check each section before and after and between when x = 0 to see if the y value of the equation is negative or positive.
pick a value for x and solve the equation using that value.
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assume x = -2 which is less than -5/3.
when x = -2, the expression of 6x * (x-2) * (3x+5) = -12*(-4)*(-1) which will be negative because you have a minus times a minus times a minus.
this means the equation is negative before the value of x = -5/3.
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assume x = -1 which is greater than -5/3 but less than 0.
when x = -1, the expression of 6x * (x-2) * (3x+5) = -6*(-3)*2) = 36 which is a minus times a minus times a plus.
this means that the equation is positive between the values of x = -5/3 and x = 0.
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assume x = 1 which is greater than x = 0 and less than x = 2.
when x = 1, the expression of 6x * (x-2) * (3x+5) = 6*(-1)*(15) which is a plus times a minus times a plus.
this means that the equation is negative between the values of x = 0 and x = 2.
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assume x = 3 which is greater than x = 2.
when x = 3, the expression of 6x * (x-2) * (3x+5) = 18*1*14 which is a plus times a plus times a plus.
this means that the equation is positive after the value of x = 2.
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put these all together and you get:
when x < -5/3, the equation is negative.
when x > -5/3 but < 0, the equation is positive
when x > 0 but < 2, the equation is negative.
when x > 2, the equation is positive.
the question is to find when the equation is negative.
that occurs when:"
x < -5/3
x > 0 but < 2
all other times the equation is either equal to 0 or greater than 0.
this can be written as:
x < -5/3 or 0 < x < 2
check the graph again to see if it concurs.
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