Problem 112 - A Graphical Approach to Precalculus with Limits

Question

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x^{2}-x+3<0\) (b) \(2 x^{2}-x+3 \geq 0\)

Step-by-Step Solution

Answer

(a) No solution (empty set); (b) Solution is all real numbers.

Step 1: Recognize the Inequalities

The exercise involves solving two quadratic inequalities separately: (a) \(2x^2 - x + 3 < 0\) and (b) \(2x^2 - x + 3 \geq 0\). The expression \(2x^2 - x + 3\) defines a quadratic function, and we need to determine intervals where this function satisfies each inequality.

Step 2: Find the Roots of the Quadratic Equation

First, find the roots of the quadratic equation \(2x^2 - x + 3 = 0\) to identify potential intervals. Use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 2\), \(b = -1\), and \(c = 3\). Calculate the discriminant:\(b^2 - 4ac = (-1)^2 - 4(2)(3) = 1 - 24 = -23\). Since the discriminant is negative, there are no real roots.

Step 3: Analyze the Sign of the Quadratic

Since the quadratic \(2x^2 - x + 3\) has no real roots (due to a negative discriminant), the function does not intersect the x-axis. The parabola opens upwards (because \(a = 2 > 0\)). Thus, the entire parabola is above the x-axis.

Step 4: Solve Each Inequality

- For (a) \(2x^2 - x + 3 < 0\):Since the parabola does not intersect nor go below the x-axis (being entirely above), there are no real values of \(x\) where the inequality holds. Thus, the solution is the empty set.- For (b) \(2x^2 - x + 3 \geq 0\):Since the entire parabola is above the x-axis, the inequality holds for all \(x\). Therefore, the solution is \((-\infty, \infty)\).

Step 5: Graphical Representation

Graph the function \(f(x) = 2x^2 - x + 3\). Since it is a parabola opening upwards with no x-axis intersections, it remains entirely above the x-axis. Visually, this confirms:- No part of the parabola is below the x-axis for inequality (a).- The entire parabola is above or touching the x-axis for inequality (b).

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool that helps us find the roots of any quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \). When you need to solve for \( x \), the quadratic formula comes into play:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation. The symbol \( \pm \) indicates that there are generally two solutions, one with addition and one with subtraction.
For the exercise given:

• \( a = 2 \)
• \( b = -1 \)
• \( c = 3 \)

calculate the discriminant \( b^2 - 4ac \) to guide the next steps. A positive discriminant means the equation has two real roots, zero means one real root, and a negative discriminant means no real roots—only complex ones. This case has a negative discriminant.

Graphical Solutions

Graphical solutions provide a visual representation of quadratic functions, which can offer deep insight into their behavior. By graphing the function \( f(x) = 2x^2 - x + 3 \), we can observe its shape and position relative to the x-axis. Parabolas have a distinctive U-shape, and in this instance, the quadratic term \( 2x^2 \) determines the concavity.
Since \( a = 2 \) is positive, the parabola opens upwards. The roots of the equation, if real, would indicate where the parabola crosses the x-axis. However, no real roots are present here, which means the quadratic does not cross or touch the x-axis.
By observing this graph:

• All parts of the parabola are above the x-axis so this provides a graphic proof that \( 2x^2 - x + 3 \geq 0 \) for all \( x \).
• This confirms that \( 2x^2 - x + 3 < 0 \) is not possible, shown graphically as the parabola never dips below the x-axis.

Discriminant Analysis

Discriminant analysis is a key step in understanding the roots of a quadratic equation. The discriminant is given by \( b^2 - 4ac \) and it tells us how many and what type of roots the quadratic equation will have.
In this exercise, the discriminant is calculated as:

• \( (-1)^2 - 4(2)(3) = 1 - 24 = -23 \)

A negative discriminant, such as the \(-23\) found here, implies the quadratic equation has no real roots, meaning the graph of the parabola does not intersect the x-axis at any point. Instead, it suggests complex roots which are not visually represented on the standard coordinate system. This is crucial when solving inequalities analytically as it dictates the nature of intervals that satisfy or fail the inequality conditions. Discriminant analysis simplifies determining the presence or absence of real solutions.

Quadratic Functions

Quadratic functions are polynomial functions that graph as parabolas. They are expressed in the form \( f(x) = ax^2 + bx + c \). Each parabolic curve has distinct characteristics based on its coefficients:

• The coefficient \( a \) affects the direction of the opening; when \( a > 0 \), it opens upwards.
• The value of \( b \) can influence the vertex, the highest or lowest point of the parabola.
• The constant \( c \) determines the y-intercept, where the graph crosses the y-axis.

In solving inequalities like \( 2x^2 - x + 3 < 0 \) or \( 2x^2 - x + 3 \geq 0 \), we are often interested in the function's intervals. What makes quadratic functions interesting is how changes in \( a \), \( b \), or \( c \) shift and shape the parabola. The lack of real roots in this scenario implies our function resides entirely above the x-axis, meaning it is always non-negative, critical in understanding the nature of the inequalities involved.