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Chapter 2: Problem 18

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

Short Answer

Expert verified
The solution for the inequality \(4x^2 + 1 \geq 4x\) are all real numbers, given in interval notation as \(-\infty, \infty\).

Step by step solution

01

Rearranging the inequality

First, let's rearrange the given inequality \(4x^2 + 1 \geq 4x\) into standard quadratic form by bringing all terms to one side. This results in \(4x^2 - 4x + 1 \geq 0\).
02

Factoring the quadratic expression

Next, let's factorize the quadratic expression \(4x^2 - 4x + 1\). However, the expression is already in standard form and cannot be factored.
03

Identify the roots

Solving for the roots of the quadratic equation \(4x^2 - 4x + 1 = 0\), we use the formula \(-\frac{b}{2a}\) (since the discriminant \(b^2-4ac\) is less than 0 for this equation, indicating no real roots). Plugging the values, \(a=4\) and \(b=-4\), we have \(\frac{-(-4)}{2 * 4}\) = 0.5. This is the vertex of the parabola and since the coefficient of \(x^2\) is positive, the parabola opens upwards, indicating that the function is always greater than 0 for all real x.
04

Graph and determine intervals

Graph the inequality (a vertically opening parabola with vertex at x=0.5). This will indicate that the function is greater than 0 for all real x. Therefore, the solution is all real numbers.
05

Express in interval notation

Since the solution includes all real numbers, the interval notation is \(-\infty, \infty\)

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Most popular questions from this chapter

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{3 x+7}{x+2}$$

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=3,\) a horizontal asymptote \(y=0, y\) -intercept at \(-1,\) and no \(x\) -intercept.

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+4}{x}$$

Solve each inequality using a graphing utility. $$\frac{1}{x+1} \leq \frac{2}{x+4}$$
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