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Chapter 2: Problem 28

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$4 x^{2}-4 x+1 \geq 0$$

Short Answer

Expert verified
The solution to the inequality is \((-\infty, \infty)\)

Step by step solution

01

Factorize the inequality

We have the quadratic inequality \(4x^2 - 4x + 1 \geq 0\). This can be rewritten as \(4(x^2 - x + \frac{1}{4}) \geq 0\), then factored to give \((2x-1)^2 \geq 0\)
02

Find the roots

To find the roots of the inequality, we set \((2x-1)^2 = 0\). This gives \(x=\frac{1}{2}\). The root divides the number line into two intervals, \(-\infty < x < \frac{1}{2}\) and \(\frac{1}{2} < x < \infty\)
03

Test each interval

Select any value from each interval, say \(x=0\) and \(x=1\), and substitute into the inequality. We obtain \((2*0-1)^2 \geq 0\) and \((2*1-1)^2 \geq 0\), which are both true. Therefore each interval meets the inequality criteria.
04

Express the solution in interval notation

The solution to the inequality includes all real numbers, expressed in interval notation as \((-\infty, \infty)\)

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm solving a polynomial inequality that has a value for which the polynomial function is undefined.

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

What is a rational function?

Solve each inequality using a graphing utility. $$\frac{x+2}{x-3} \leq 2$$

Will help you prepare for the material covered in the next section. Simplify: \(\frac{x+1}{x+3}-2\)
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