Question

Use a graph to find a number \(N\) such that if \(x > N\) then \(\left| {\frac{{3{x^2} + 1}}{{2{x^2} + x + 1}} - 1.5} \right| < 0.05\).

Short answer

Thus, the required answer is \(N = 15\).

Step-by-step solution

Rewrite the given inequality

Apply the absolute property, that is, if \(\left| x \right| < a\) then \( - a < x < a\).

Rewrite the given condition \(\left| {\frac{{3{x^2} + 1}}{{2{x^2} + x + 1}} - 1.5} \right| < 0.05\) as:

\(\begin{aligned} - 0.05 &< \frac{{3{x^2} + 1}}{{2{x^2} + x + 1}} - 1.5 < 0.05\\ - 0.05 + 1.5 &< \frac{{3{x^2} + 1}}{{2{x^2} + x + 1}} < 0.05 + 1.5\\1.45 &< \frac{{3{x^2} + 1}}{{2{x^2} + x + 1}} &< 1.55\end{aligned}\)

Plot the graph of the function

Draw the graph of the function\(f\left( x \right) = \frac{{3{x^2} + 1}}{{2{x^2} + x + 1}}\),\(g\left( x \right) = 1.45\) and \(h\left( x \right) = 1.55\) by using the graphing calculator as:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\frac{{3{x^2} + 1}}{{2{x^2} + x + 1}}\)in the\({Y_1}\)tab,\(1.45\)in the \({Y_2}\)tab and\(1.55\) in the \({Y_3}\)tab.

2. Enter the “GRAPH” button in the graphing calculator.

3. Use “INTERSECT” feature of graphing to obtain the solution.

The graph of the functions is shown below:

Observe the graph

From the graph, it is observed that the curve of the function crosses the line \(y = 1.45\) when \(x \approx 14.8\).

This implies that the curve lies between \(1.45\) and \(1.55\) when \(x \approx 15\).

Thus, if \(x > 15\) then \(\left| {\frac{{3{x^2} + 1}}{{2{x^2} + x + 1}} - 1.5} \right| < 0.05\).