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Chapter 2: Problem 29

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$(x-1)(x-2)(x-3) \geq 0$$

Short Answer

Expert verified
The solution to the inequality \((x-1)(x-2)(x-3) \geq 0\) is \((-\inf, 1], (2, 3]\) and \([3, \inf)\).

Step by step solution

01

Find the roots of the polynomial

This step involves identifying the roots of the equation where it equals zero, which are also the boundary points of our intervals. Given the equation \((x-1)(x-2)(x-3) \geq 0\), the roots are easily identified by setting each expression to zero. Thus we get \(x=1\), \(x=2\), and \(x=3\). These are points where the polynomial changes its sign.
02

Dividing the number line into intervals

We'll divide the number line using the roots from step 1. The roots split the number line into four intervals: \(-\inf < x < 1\), \(1 < x < 2\), \(2 < x < 3\), and \(3 < x < \inf\). Each of these intervals will be tested in the next step.
03

Test each interval

Testing determines whether the polynomial is positive or negative over each interval. Choose a test point in each interval and substitute it into the inequality. If the result is positive, the inequality is satisfied for that interval. If the result is negative, the inequality is not satisfied. For example, use \(x = 0\) for the first interval. \((0 - 1)(0 - 2)(0 - 3)\) yields a positive result, so the first interval satisfies the inequality.
04

Write the solution in interval notation and graph it on a number line

For every interval that satisfies the inequality, write it in interval notation. In this example, the solutions to the inequality are \((-\inf, 1], (2, 3]\) and \([3, \inf)\). The graph should clearly mark each of the intervals where the polynomial is either positive or zero. Notice that, because the inequality is \(\geq\), the solution includes the roots of the polynomial, which splits the number line. This is represented in interval notation by using square brackets \([]\) instead of parentheses \(()\) to include the end points.

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Most popular questions from this chapter

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$(x-2)^{2}>0$$

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x-2}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using the language of variation, I can now state the formula for the area of a trapezoid, \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right),\) as, "A trapezoid's area varies jointly with its height and the sum of its bases."

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$(x)=\frac{1}{(x+2)^{2}}$$

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+2$$
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