Solving Quadratic Inequalities: A Step-by-Step Guide

Let's dive into solving the quadratic inequality $x^2 + 5x - 26 ">=" x - 5$. This type of problem appears frequently in mathematics, and understanding the process to solve it is fundamental. This article will guide you through each step, ensuring you grasp the underlying concepts and can confidently tackle similar problems. We will break down the process into manageable parts: simplifying the inequality, finding critical points, testing intervals, and expressing the solution set. By the end of this guide, you’ll not only know how to solve this specific inequality but also have a solid understanding of the methodology applicable to a broad range of quadratic inequalities.

1. Simplify the Inequality

First, let’s simplify the given inequality: $x^2 + 5x - 26 ">=" x - 5$. Our goal is to get all terms on one side, leaving zero on the other side. This standard form makes it easier to analyze the quadratic expression. Subtract $x$ from both sides to get $x^2 + 4x - 26 ">=" -5$. Now, add $5$ to both sides to obtain $x^2 + 4x - 21 ">=" 0$. We now have a standard quadratic inequality ready for factoring or further analysis. Having a zero on one side allows us to focus on finding the roots of the quadratic expression and determining the intervals where the inequality holds true. This step is crucial as it sets the stage for finding the critical points that define the boundaries of our solution set. Making sure the inequality is correctly simplified avoids errors in the subsequent steps and helps in visualizing the solution more clearly.

2. Find the Critical Points

Next, we need to find the critical points of the inequality $x^2 + 4x - 21 ">=" 0$. Critical points are the values of $x$ where the quadratic expression equals zero. These points divide the number line into intervals that we will later test. To find these points, we solve the equation $x^2 + 4x - 21 = 0$. This can be done by factoring, completing the square, or using the quadratic formula. In this case, factoring is straightforward. We look for two numbers that multiply to $-21$ and add to $4$. These numbers are $7$ and $-3$. Thus, we can factor the quadratic expression as $(x + 7)(x - 3) = 0$. Setting each factor to zero gives us the critical points: $x + 7 = 0$ implies $x = -7$, and $x - 3 = 0$ implies $x = 3$. These two values, $-7$ and $3$, are the critical points that will determine the intervals we need to test. They are the roots of the quadratic equation and are essential in determining where the quadratic expression changes its sign. Accurately identifying these critical points is vital for correctly solving the inequality.

3. Test Intervals

Now that we have the critical points $x = -7$ and $x = 3$, we will test the intervals they define to determine where the inequality $x^2 + 4x - 21 ">=" 0$ holds true. The critical points divide the number line into three intervals: $(-\infty, -7]$, $[-7, 3]$, and $[3, \infty)$. We need to pick a test value from each interval and plug it into the inequality to see if it satisfies the condition.

• Interval 1: $(-\infty, -7]$

  Let's pick $x = -8$. Plugging this into the inequality, we get $(-8)^2 + 4(-8) - 21 = 64 - 32 - 21 = 11$. Since $11 ">=" 0$, the inequality holds true in this interval.

• Interval 2: $[-7, 3]$

  Let's pick $x = 0$. Plugging this into the inequality, we get $(0)^2 + 4(0) - 21 = -21$. Since $-21 < 0$, the inequality does not hold true in this interval.

• Interval 3: $[3, \infty)$

  Let's pick $x = 4$. Plugging this into the inequality, we get $(4)^2 + 4(4) - 21 = 16 + 16 - 21 = 11$. Since $11 ">=" 0$, the inequality holds true in this interval.

Thus, the inequality holds true for the intervals $(-\infty, -7]$ and $[3, \infty)$. Testing intervals is a crucial step in solving inequalities. By choosing representative values from each interval, we can determine the sign of the quadratic expression in that interval and thus identify the regions where the inequality is satisfied. This process ensures that we capture all the solutions to the inequality and accurately define the solution set.

4. Express the Solution Set

Finally, we express the solution set for the inequality $x^2 + 5x - 26 ">=" x - 5$, which we simplified to $x^2 + 4x - 21 ">=" 0$. From our interval testing, we found that the inequality holds true for the intervals $(-\infty, -7]$ and $[3, \infty)$. Therefore, the solution set is the union of these two intervals. In interval notation, this is written as $(-\infty, -7] \cup [3, \infty)$. This notation indicates that any value of $x$ less than or equal to $-7$ or greater than or equal to $3$ will satisfy the original inequality. The square brackets indicate that the endpoints $-7$ and $3$ are included in the solution set because the inequality is non-strict (i.e., it includes the "equal to" case). Writing the solution set correctly is important for providing a complete and accurate answer. This final step ties together all the previous steps and presents the solution in a clear and concise manner.

Conclusion

In summary, solving the quadratic inequality $x^2 + 5x - 26 ">=" x - 5$ involves several key steps: simplifying the inequality to standard form, finding the critical points by solving the corresponding quadratic equation, testing intervals defined by these critical points, and expressing the solution set using interval notation. Following these steps meticulously ensures an accurate solution. Remember, practice is key to mastering these skills. Work through various examples to reinforce your understanding and build confidence in solving quadratic inequalities. This methodical approach not only helps in solving specific problems but also enhances your overall problem-solving abilities in mathematics. Understanding these concepts thoroughly will be invaluable for more advanced topics in algebra and calculus. Don't hesitate to review each step and seek additional resources if needed. With consistent effort and practice, you can confidently tackle any quadratic inequality that comes your way.

For further reading and more examples, you can visit Khan Academy's section on quadratic inequalities: Khan Academy - Quadratic Inequalities.