Problem 108 - Algebra and Trigonometry Real Mathematics, Real People

Question

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.

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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: \(D = (3)^{2} - 4(\frac{3}{2})(6) = 9 - 36 = -27\).

Step 3: 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.

Step 4: 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.