I have done one part of this question. I am, however, unsure as to how to set up the graph. Can someone help me?

1. Use a graph to find a number N such that

|(6x^2+5x-3)| |
|------------- -3 | < 0.2 whenever x>N
|(2x^2-1)| |

y=2.8 and y=3.2

My issue is how to find the x, or where the curve crosses at the horizontal line at 2.8.

the equation is supposed to be

|(6x^2+5x-3)/(2x^2-1) -3 (the negative three is outside of the fraction) |

let y = (6x^2 + 5x - 3)/(2x^2 - 1) - 3

we can simplify
= ( 6x^2 + 5x - 3 - 3(2x^2 - 1) )/(2x^2 - 1)
= (5x)/(2x^2 - 1)

there will be 2 vertical asymptotes,
x = ± 1/√2

graph looks like this, the asymptotes are not drawn
http://www.wolframalpha.com/input/?i=simplify+%286x%5E2+%2B+5x+-+3%29%2F%282x%5E2+-+1%29+-+3

let's set 5x/(2x^2 - 1) = .2 = 1/5
25x = 2x^2 - 1
2x^2 - 25x - 1 = 0
x = (25 ± √633)/4
= appr 12.54 or -.04

( I don't know how you got your answers)

so (6x^2+5x-3)/(2x^2-1) -3 < .2
for all x>12.54 or x < -.04

To set up the graph and find the value of N, you can follow these steps:

1. Simplify the equation:
|(6x^2+5x-3)| / |(2x^2-1)| - 3 < 0.2

2. Rearrange the equation:
|(6x^2+5x-3)| / |(2x^2-1)| < 3.2

3. Notice that the equation involves absolute values, so break it down into two cases:
Case 1: (6x^2+5x-3) / (2x^2-1) < 3.2
Case 2: (6x^2+5x-3) / (2x^2-1) > -3.2

4. Simplify each case separately. Let's start with Case 1:
(6x^2+5x-3) / (2x^2-1) < 3.2

5. Set the numerator equal to y (the horizontal line) and solve for x:
6x^2 + 5x - 3 = y

6. Use the specific y-values given in the question (y=2.8 and y=3.2) to find the corresponding x-values by solving the quadratic equation (6x^2 + 5x - 3 = y).

7. Plot the two points (x,y) obtained from solving the equation in step 6 on a graph.

8. Repeat steps 4-7 for Case 2 (using the inequality (6x^2+5x-3) / (2x^2-1) > -3.2) to find two more points on the graph.

9. Connect the points obtained in steps 7 and 8 with a smooth curve.

10. The value of N is the x-coordinate of the point where the curve crosses the horizontal line at y=2.8.