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Chapter 3: Problem 10

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

Short Answer

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

Step by step solution

01

Factor the polynomial

The given polynomial inequality is a perfect square trinomial, it can be factored as \((x-1)^2>0\)
02

Solve for x

We set the equation \((x-1)^2=0\). The solution to this equation, also called the critical number, is \(x=1\).
03

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.
04

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 \((1, \infty)\)

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Most popular questions from this chapter

A mong all deaths from a particular disease, the percentage that is smoking related \((21-39\) cigarethes per day) is a function of the discase's incidence ratio. The incidence ratio describes the number of times more likely smokers are than nonsmokers to die from the disease. The following table shows the incidence ratios for heart disease and lung cancer for two age groups. For example, the incidence ratio of 9 in the table means that smokers befween the ages of 65 and 74 are 9 times more likely than nonsmokers in the same age group to die from lung cancer. The rational function$$ P(x)=\frac{100(x-1)}{x} $$models the percentage of smoking-related deaths among all deaths from a disease, \(P(x),\) in terms of the disease's incidence ratio, \(x\). The graph of the rational function is shown. What is the horizontal asymptote of the graph? Describe what this means about the percentage of deaths caused by smoking with increasing incidence ratios.

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Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the ZOOM OUT feature to show that f and g have identical end behavior. \(f(x)=-x^{4}+2 x^{3}-6 x, \quad g(x)=-x^{4}\)

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