Problem 16 - Algebra 2 and Trigonometry

Question

In \(12-17,\) use a graph to find the solution set of each inequality. $$ -x^{2}+6 x-5 < 0 $$

Step-by-Step Solution

Answer

The solution set is \(x < 1\) or \(x > 5\).

Step 1: Understand the Inequality

The inequality given is \(-x^2 + 6x - 5 < 0\). It represents a parabola opening downwards, as the coefficient of \(x^2\) is negative.

Step 2: Rewrite the Inequality as an Equation

First, treat the inequality as an equation: \(-x^2 + 6x - 5 = 0\). Solving this will help us find the points where the parabola intersects the x-axis, which are the boundary points.

Step 3: Solve the Quadratic Equation

To find the roots, use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = -1\), \(b = 6\), and \(c = -5\). Calculate the discriminant: \[ b^2 - 4ac = 6^2 - 4(-1)(-5) = 36 - 20 = 16 \] The roots are: \[ x = \frac{-6 \pm \sqrt{16}}{-2} \] This simplifies to: \[ x = \frac{-6 \pm 4}{-2} \] Giving the solutions: \(x = 1\) and \(x = 5\).

Step 4: Plot the Graph

Plot the quadratic function \(-x^2 + 6x - 5\) on a graph. The roots \(x = 1\) and \(x = 5\) mean that the parabola crosses the x-axis at these points. Since the parabola opens downwards, it is above the x-axis between these two roots.

Step 5: Determine the Solution Set for the Inequality

Since the inequality is \(-x^2 + 6x - 5 < 0\), we are interested in the portion where the graph is below the x-axis. This occurs outside the interval between the roots. Thus, the solution set is: \(x < 1\) or \(x > 5\).

Key Concepts

Quadratic Formula

When solving quadratic equations, the quadratic formula is an immensely powerful tool that comes in handy. It can find the roots of any quadratic equation in the form ax^2 + bx + c = 0. The formula is structured as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here,

• "a" is the coefficient of x^2,
• "b" is the coefficient of x, and
• "c" is the constant term.

The term under the square root sign, \( b^2 - 4ac \), is known as the discriminant. It plays a crucial role in determining the nature of the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there's exactly one real root, meaning the parabola touches the x-axis at a single point.
• If it's negative, there are no real roots, but rather two complex roots.

In our example, we have a downward-opening parabola given by the equation \(-x^2 + 6x - 5 = 0\). Plugging in the values \(a = -1\), \(b = 6\), and \(c = -5\) into the quadratic formula, we find the roots as \(x = 1\) and \(x = 5\). These roots indicate where the curve intersects the x-axis. This helps in identifying the regions that satisfy the given inequality.

Graphing Inequalities

Graphing inequalities involves a visual representation where a parabola is plotted on the coordinate plane. Here, we need to determine the regions on the x-axis which will satisfy the inequality. Our inequality, \(-x^2 + 6x - 5 < 0\), denotes that we seek the points where this downward-opening parabola is positioned below the x-axis around the roots found earlier.
To graph this inequality:

• We start by graphing the related equation, \(-x^2 + 6x - 5 = 0\).
• This curve intersects the x-axis at \(x = 1\) and \(x = 5\), forming the boundary points.

Next, visualize that the segments of the x-axis outside of these intersection points (i.e., \(x < 1\) and \(x > 5\)) are where the inequality holds true. Essentially, we are identifying the regions where this parabola dips below the x-axis.
Drawing and shading these areas on the graph will illuminate the solution set more clearly. This visual understanding is powerful for grasping where the inequality is satisfied.

Parabola Intersections

Parabola intersections on a graph help identify the critical values which partition the plane into different regions. For a parabola represented by a quadratic equation, the points where it crosses the x-axis are known as its roots or zeros. In solving quadratic inequalities, these intersections denote transition points where the graph changes in relation to the x-axis.
In the given quadratic inequality \(-x^2 + 6x - 5 < 0\), the intersections at \(x = 1\) and \(x = 5\) form key boundaries:

• Within the interval \(1 < x < 5\), the parabola is positioned above the x-axis, suggesting that this region does not satisfy the inequality.
• In contrast, for \(x < 1\) and \(x > 5\), the parabola dips below the x-axis, meaning the inequality \(-x^2 + 6x - 5 < 0\) holds true.

Understanding where a parabola intersects the x-axis helps us determine which sections are part of the solution set when dealing with inequalities. It's a simple yet profound tool that enhances both algebraic and geometric reasoning.