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Chapter 3: Problem 18

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. $$ 4 x^{2}+1 \geq 4 x $$

Short Answer

Expert verified
The solution set of the inequality is all real numbers represented in interval notation as \((- \infty, \infty)\).

Step by step solution

01

Rearrange the Inequality

Subtract \( 4x \) from both sides to get the inequality in the form \( 4x^2-4x+1\geq 0 \). This makes it easier to identify the terms and to plot on the number line.
02

Solving the Quadratic Inequality

Set up a quadratic equation. To find the roots of the equation \(4x^2 - 4x + 1 = 0 \) we can use quadratic formula which is \(x = -b \pm \sqrt{b^2-4ac} / (2a)\) where \(a = 4\), \(b = -4\), and \(c = 1\). Substituting these variables into the quadratic formula, we find that the roots are complex, indicating that the quadratic does not intersect the x-axis.
03

Determine the Solution Set in Interval Notation

Since the roots are complex, it means that the quadratic \(4x^2 - 4x + 1\) is always positive or always negative. To find out, we check the coefficient of the leading term (the term with the highest power of x), which in this case is 4 and it is positive. It indicates that the parabola opens upwards. Also, the discriminant (\(b^2 - 4ac\)) is negative, so the vertex of the parabola is above the x-axis. Therefore, the inequality \(4x^2 -4x + 1 \geq 0\) is true for all real numbers.
04

Graph the Solution on a Number Line

Since the inequality holds true for all real numbers, we will have a solid line extending infinitely to the left and right.

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Most popular questions from this chapter

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=6 x-x^{3}-x^{5}$$

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 \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{2 x-9}{x-4} $$

Explain how to use the Leading Coefficient Test to determine the end behavior of a polynomial function.

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