Question

Consider the following rational inequality and answer the following questions:  3/2x-1 < -1

Move the -1 to the left side of the inequality

Take the denominator and multiply by the whole number and combine with the numerator

Simplify the numerator

Solve for x to find the x-intercepts and y-intercepts as ordered pairs.

Find the vertical asymptote and the horizontal asymptote

Answer 1

3/(2x-1)<-1,

If 2x-1>0, x>½:

3<-2x+1, 2x+2<0, x<-1, in contradiction to x>½.

Therefore 2x-1<0, so x<½.

When x=½, we have

3/(2x-1)<-1,

If 2x-1>0, x>½:

3<-2x+1, 2x+2<0, x<-1, in contradiction to x>½.

Therefore 2x-1<0, so x<½.

When x=½, we have a vertical asymptote.

Also, 3>-2x+1 because -2x+1 is negative.

And 2x+2>0, x>-1.

Therefore, -1<x<½ satisfies both conditions.

When x is very large  and y=3/(2x-1)+1, the horizontal asymptote is y=1.