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Chapter 2: Problem 10

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

Short Answer

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

Step by step solution

01

Solve The Quadratic Equation

Let's first solve the quadratic equation \(x^{2}-2 x+1=0\). This Quadratic is a perfect square trinomial. So it can be factored and set equal to zero to find the root(s): \((x-1)^{2} = 0\), which leads us to \(x = 1\).
02

Find the Test Intervals

The solution \(x = 1\) divides the number line into two intervals: \(-\infty, 1\) and \(1, \infty\). Each interval then is tested to see if it satisfies the original inequality.
03

Test Each Interval

Let's randomly choose test numbers from each interval. For \(-\infty, 1\), let's take \(x = 0\). Plugging back into the inequality: \((0)^2 - 2*(0) + 1 > 0\), we get \(1 > 0\), which is true. So, the interval \(-\infty, 1\) satisfies the inequality. Next, let's take \(x = 2\) from the interval \(1, \infty\). \(2^{2} - 2*2 + 1 > 0\), we get \(1 > 0\), which is true. So, the interval \(1, \infty\) also satisfies the inequality.
04

Express the Solution in Interval Notation and Graphically

The solution is all real numbers except \(x = 1\). In interval notation, the answer would be \((- \infty,1) \cup (1, \infty)\). The graphic representation shows two rays on the number line, one going from 1 to negative infinity, not including 1, and the other from 1 to positive infinity, not including 1.

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