Problem 40 - Algebra and Trigonometry

Question

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}-x^{2}+9 x-9>0 $$

Step-by-Step Solution

Answer

The solution set in interval notation is \((1.0047, +\infty)\)

Step 1: Factorize the cubic equation

The first step is to factorize the equation \(x^{3}-x^{2}+9x -9 = 0\). However, this specific polynomial doesn't have a simple factorization. Hence, we can use the Rational Root Theorem or approximations to find its roots.

Step 2: Find the roots

The root of this cubic equation can be found by graphical analysis or using a calculator. In this case, the roots of the equation \(x^{3}-x^{2}+9x -9 = 0\) are approximately \(x \approx 1.0047\), \(x \approx -3.3971-0.7104i\), and \(x \approx -3.3971+0.7104i\). Considering we're graphing on a real number line, we only need the real root, which is \(x \approx 1.0047\).

Step 3: Determine the intervals

We can now establish intervals for the inequality. We have one real root, dividing the line into two intervals. These are \((-\infty, 1.0047)\) and \((1.0047, +\infty)\). Choose a test point in each interval to check if the inequality is satisfied.

Step 4: Check the intervals

We need to substitute the test points into the inequality. For the first interval, let's pick \(x=0\). Substituting that into \(x^{3}-x^{2}+9x -9 > 0\), we find that it's less than zero, therefore this interval isn't part of the solution set. For the second interval, let's pick \(x=2\), doing the same we find that it's greater than zero, therefore this interval is part of the solution set.

Step 5: Express in interval notation

Finally, express the solution as an interval, taking into account the inequality is 'greater than' and not 'greater than or equal to'. This indicates that we should use a parenthesis, not square brackets. Our solution in interval notation becomes \((1.0047, +\infty)\).