Simplify the inequality, identify any critical points, and graph its solution.  (1-4x)/(2x^2+3x-2)+(2x)/(x+2)≤(3x)/(2x-1) Refer to the photo to see the number line 

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Solve the inequality and graph the solution:

`(1-4x)/(2x^2+3x-2)+(2x)/(x+2) <= (3x)/(2x-1) `

`(1-4x)/((2x-1)(x+2))+(2x(2x-1))/((2x-1)(x+2))<=(3x(x+2))/((2x-1)(x+2)) `

`(4x^2-6x+1)/((2x-1)(x+2))<= (3x^2+6x)/((2x-1)(x+2)) `

`(x^2-12x+1)/((2x-1)(x+2))<=0 `

Note that the denominator is zero at x=-2 and x=1/2. The numerator is zero at `x=6 pm sqrt(35) `

So we have to check the original inequality for the following intervals:

x<-2 not true

`-2<x<=6-sqrt(35) `  true

`6-sqrt(35)<x<1/2 `  not true

`1/2<x<= 6+sqrt(35) `  true

`x>6+sqrt(35) `  not true

In interval notation the solution set is:

`(-2,6-sqrt(35)]uu(1/2,6+sqrt(35)] `

The graph of the solution:

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The graph of the left side in black, the graph of the right side in red:

Sources:

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