Problem 10 - 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. $$ x^{2}-2 x+1>0 $$

Step-by-Step Solution

Answer

The solution set for the inequality \(x^{2}-2 x+1>0\) is \((1, \infty)\)

Step 1: Factor the polynomial

The given polynomial inequality is a perfect square trinomial, it can be factored as \((x-1)^2>0\)

Step 2: Solve for x

We set the equation \((x-1)^2=0\). The solution to this equation, also called the critical number, is \(x=1\).

Step 3: Test the intervals

Now we set up test points in each of the intervals determined by the critical number, that is \(-\infty\) to \(1\) and \(1\) to \(\infty\). If we select \(0\) as a test point for the interval \(-\infty\) to \(1\) and substitute it into the inequality \((x-1)^2>0\), we observe that the inequality holds false. Therefore, the interval \(-\infty\) to \(1\) is not part of the solution. If we select \(2\) as a test point for the interval \(1\) to \(\infty\) and substitute it into the inequality, we observe that the inequality holds true. Therefore, the interval \(1\) to \(\infty\) is part of the solution.

Step 4: Graph the solution and express in interval notation

On the number line, we place an open circle at \(x=1\) to indicate that \(1\) is not included in the solution and draw an arrow to the right of \(x=1\) to indicate that all real numbers greater than \(1\) are included in the solution. The solution in interval notation is \((1, \infty)\)