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Chapter 2: Problem 108

Determine whether the statement is true or false. Justify your answer. The solution set of the inequality \(\frac{3}{2} x^{2}+3 x+6 \geq 0\) is the entire set of real numbers.

Short Answer

Expert verified
The statement is true. The solution set of the inequality \(\frac{3}{2} x^{2}+3 x+6 \geq 0\) is indeed the entire set of real numbers.

Step by step solution

01

Identify the Quadratic Function

The quadratic function in this exercise is \(f(x) = \frac{3}{2} x^{2} + 3x + 6\). This function opens upwards because the coefficient of \(x^{2}\) is positive.
02

Find the Discriminant

The Discriminant, often denoted by \(D\), in a quadratic equation, is calculated as \(D = b^{2} - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of \(x^{2}\), \(x\), and the constant term respectively. So here, \(a = \frac{3}{2}\), \(b = 3\), and \(c = 6\). Substituting these into the discriminant formula gives: \(D = (3)^{2} - 4(\frac{3}{2})(6) = 9 - 36 = -27\).
03

Analyze Discriminant

A negative discriminant denotes that there are no real roots for the quadratic function. It also means that the parabola represented by this function will not cut or touch the \(x\)-axis. The lowest point in an upward-opening parabola is its vertex. Since the function does not touch the \(x\)-axis, the y-coordinate of the vertex will be greater than 0.
04

Validate the Statement

Now, because the function is always above the x-axis (greater than or equal to 0), for all values of \(x\), the solution set of the inequality is indeed the entire set of real numbers. Therefore, the given statement is true.

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Most popular questions from this chapter

Two planes leave simultaneously from Chicago's O'Hare Airport, one flying due north and the other due east (see figure). The northbound plane is flying 50 miles per hour faster than the eastbound plane. After 3 hours, the planes are 2440 miles apart. Find the speed of each plane.

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x^{4}-10 x^{2}+9$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=3 x-3 \sqrt{x}-4$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x-15|=x^{2}-15 x$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$\frac{4}{x+1}-\frac{3}{x+2}=1$$
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