Solving The Quadratic Inequality X² - 6x ≥ -8

In this article, we will delve into the process of solving the quadratic inequality x² - 6x ≥ -8. This type of problem is a fundamental concept in algebra and is crucial for understanding various mathematical applications. We will break down the steps involved, providing a clear and concise explanation to help you grasp the underlying principles. Quadratic inequalities, like the one we're addressing, often appear in various mathematical contexts, from calculus to optimization problems. Mastering the techniques to solve them is essential for any student or professional working with mathematical models.

Understanding Quadratic Inequalities

Before we dive into the specific problem, let's briefly discuss quadratic inequalities in general. A quadratic inequality is an inequality that involves a quadratic expression, which is a polynomial expression of degree two. The general form of a quadratic inequality is ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0, where a, b, and c are constants, and a ≠ 0. The solutions to these inequalities are the values of x that satisfy the inequality. These solutions often represent intervals on the number line, which can be determined by analyzing the roots of the quadratic equation and the sign of the quadratic expression in different intervals.

Solving quadratic inequalities involves several key steps. First, we need to rewrite the inequality so that one side is zero. This allows us to identify the critical points, which are the roots of the corresponding quadratic equation. These critical points divide the number line into intervals, and we can then test a value from each interval to determine where the inequality holds true. The process of solving quadratic inequalities is not just a matter of following a set of rules; it requires a solid understanding of how quadratic functions behave and how their graphs relate to the solutions of inequalities. This conceptual understanding is what we aim to reinforce throughout this article.

By the end of this discussion, you should have a solid understanding of how to solve quadratic inequalities like x² - 6x ≥ -8. We will cover each step in detail, ensuring you're well-equipped to tackle similar problems in the future. This involves not only the mechanical steps of solving the inequality but also understanding the underlying concepts that make the solution process logical and intuitive. Our goal is to provide a comprehensive guide that serves as a valuable resource for anyone looking to master this important mathematical skill.

Step 1: Rewrite the Inequality

The first step in solving the inequality x² - 6x ≥ -8 is to rewrite it so that one side is zero. This is crucial because it allows us to easily identify the critical points, which are the roots of the corresponding quadratic equation. To do this, we add 8 to both sides of the inequality:

x² - 6x + 8 ≥ 0

Now, we have a quadratic expression on the left side and zero on the right side. This form is essential for the next steps in solving the inequality. By setting the inequality to zero, we create a standard form that facilitates the identification of roots and the analysis of intervals. This step is not just about algebraic manipulation; it's about transforming the problem into a format that is more amenable to the methods we will use to find the solution.

This rewritten form, x² - 6x + 8 ≥ 0, sets the stage for the subsequent steps, which involve factoring the quadratic expression and finding the critical points. Without this initial step, it would be significantly more challenging to determine the intervals where the inequality holds true. The importance of this step cannot be overstated, as it forms the foundation for the rest of the solution process. The algebraic manipulation we perform here is a common technique in solving inequalities and equations, and mastering it is crucial for success in algebra and beyond.

Furthermore, rewriting the inequality in this form allows us to visualize the problem graphically. The quadratic expression x² - 6x + 8 represents a parabola, and the inequality x² - 6x + 8 ≥ 0 asks us to find the values of x for which the parabola is above or on the x-axis. This graphical interpretation provides a valuable perspective on the problem and helps to solidify our understanding of the solution. The connection between algebraic manipulation and graphical representation is a key aspect of mathematical problem-solving, and this step exemplifies that connection perfectly.

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression x² - 6x + 8. Factoring allows us to find the roots of the corresponding quadratic equation, which are the critical points for our inequality. We are looking for two numbers that multiply to 8 and add up to -6. These numbers are -2 and -4. Therefore, we can factor the expression as follows:

(x - 2)(x - 4) ≥ 0

Now we have the quadratic expression in factored form. This form is incredibly useful because it allows us to easily identify the values of x that make the expression equal to zero. These values are the roots of the equation (x - 2)(x - 4) = 0, which are x = 2 and x = 4. These roots are the critical points that will divide the number line into intervals that we will test in the next step. Factoring is a fundamental skill in algebra, and it's essential for solving quadratic equations and inequalities efficiently.

The process of factoring a quadratic expression involves reversing the process of expansion. In other words, we're trying to find two binomials that, when multiplied together, give us the original quadratic expression. This can sometimes be done by inspection, as in this case, or by using more systematic methods, such as the quadratic formula. However, when possible, factoring is often the most straightforward approach.

Factoring not only helps us find the roots but also gives us insight into the behavior of the quadratic expression. The factored form reveals the linear factors that make up the quadratic, and the roots are simply the values of x that make these factors equal to zero. This understanding is crucial for interpreting the solution to the inequality, as it allows us to connect the algebraic representation to the geometric interpretation of the parabola represented by the quadratic expression. The roots, in this context, are the x-intercepts of the parabola, and their position on the number line plays a key role in determining the intervals where the inequality is satisfied.

Step 3: Identify Critical Points

From the factored form (x - 2)(x - 4) ≥ 0, we can easily identify the critical points. These are the values of x that make the expression equal to zero. Setting each factor equal to zero, we get:

x - 2 = 0 => x = 2 x - 4 = 0 => x = 4

So, the critical points are x = 2 and x = 4. These points are crucial because they divide the number line into intervals where the expression (x - 2)(x - 4) is either positive or negative. These critical points are the boundaries between the regions where the inequality holds true and where it does not. Identifying these points is a key step in solving quadratic inequalities, as it sets the stage for testing the intervals and determining the solution set. The critical points are essentially the turning points of the quadratic function, where the sign of the function may change.

The critical points, x = 2 and x = 4, are the roots of the quadratic equation x² - 6x + 8 = 0. They represent the x-intercepts of the parabola described by the quadratic expression. Understanding this graphical interpretation helps to visualize the problem and the solution. The critical points act as dividers, separating the number line into intervals where the parabola is either above or below the x-axis. In the context of the inequality (x - 2)(x - 4) ≥ 0, we are looking for the intervals where the parabola is above or on the x-axis, which corresponds to the expression being greater than or equal to zero.

The process of finding critical points is a fundamental technique in solving inequalities, not just quadratic inequalities. It's a method that is used in various areas of mathematics, including calculus and optimization. The idea of identifying critical points and using them to analyze intervals is a powerful tool for understanding the behavior of functions and solving problems that involve inequalities. This step highlights the importance of algebraic manipulation in conjunction with conceptual understanding, as the critical points have both an algebraic significance (as roots of the equation) and a geometric interpretation (as x-intercepts of the parabola).

Step 4: Test Intervals

The critical points x = 2 and x = 4 divide the number line into three intervals: (-∞, 2), (2, 4), and (4, ∞). To determine where the inequality (x - 2)(x - 4) ≥ 0 holds true, we need to test a value from each interval. This testing process will reveal the sign of the expression (x - 2)(x - 4) in each interval, which will then tell us whether the inequality is satisfied in that interval. The logic behind this method is that the sign of the expression can only change at the critical points, so if the expression has a certain sign at one point in an interval, it will have the same sign throughout that interval.

Let's test each interval:

1. Interval (-∞, 2): Choose x = 0. (0 - 2)(0 - 4) = (-2)(-4) = 8 ≥ 0. The inequality holds true.
2. Interval (2, 4): Choose x = 3. (3 - 2)(3 - 4) = (1)(-1) = -1 < 0. The inequality does not hold true.
3. Interval (4, ∞): Choose x = 5. (5 - 2)(5 - 4) = (3)(1) = 3 ≥ 0. The inequality holds true.

By testing these intervals, we have determined the regions on the number line where the inequality is satisfied. This process is a systematic way of analyzing the behavior of the quadratic expression across the entire real number line. The choice of test values within each interval is arbitrary; any value within the interval will yield the same sign for the expression. The key is to select a value that is easy to work with to minimize the chances of making arithmetic errors. This step is a crucial component of solving inequalities, as it transforms the problem from an abstract algebraic inequality into a concrete analysis of intervals on the number line.

This method of testing intervals is not limited to quadratic inequalities; it can be applied to any inequality that can be expressed as a product or quotient of factors. The critical points, in general, are the values that make any of the factors zero or undefined. The intervals defined by these critical points are then tested to determine the solution set. This technique is a versatile tool in the arsenal of any mathematician or problem-solver, and it's a testament to the power of breaking down complex problems into smaller, manageable parts.

Step 5: State the Solution

Based on the interval testing, the inequality (x - 2)(x - 4) ≥ 0 holds true for the intervals (-∞, 2] and [4, ∞). Note that we include the endpoints 2 and 4 because the inequality is greater than or equal to zero. The brackets indicate that the endpoints are included in the solution set, which is a direct consequence of the