Solving Quadratic Inequalities A Step-by-Step Guide To $x^2 + X - 2 \geq 0$

Introduction

In the realm of mathematics, particularly in algebra, solving inequalities is a fundamental skill. This article delves into the process of finding the solution set for a specific quadratic inequality: $x^2 + x - 2 \geq 0$. We will explore the step-by-step methodology required to solve such inequalities, providing a comprehensive understanding that will benefit both students and math enthusiasts. Grasping the nuances of quadratic inequalities is crucial as they frequently appear in various mathematical contexts and real-world applications. Our focus will be on a clear, systematic approach, ensuring that you can confidently tackle similar problems in the future. By the end of this guide, you will not only know the solution to the given inequality but also understand the underlying principles that govern quadratic inequalities.

Step 1: Convert the Inequality to an Equation

The initial step in solving the quadratic inequality $x^2 + x - 2 \geq 0$ is to transform the inequality into an equation. This allows us to find the critical points where the quadratic expression equals zero. These critical points are crucial because they divide the number line into intervals, within which the quadratic expression will have a consistent sign (either positive or negative). By setting the inequality to an equation, we get $x^2 + x - 2 = 0$. This equation is a standard quadratic equation that we can solve using various methods, such as factoring, completing the square, or using the quadratic formula. Factoring is often the quickest and most straightforward method when the quadratic expression is factorable. The equation now provides a clear path to identifying the values of x that make the expression equal to zero, which are the foundation for determining the solution set of the original inequality. Understanding this conversion is a cornerstone in solving quadratic inequalities, as it sets the stage for the subsequent steps.

Step 2: Solve the Quadratic Equation

Having converted the quadratic inequality into the equation $x^2 + x - 2 = 0$, our next task is to solve for x. In this case, the quadratic expression is easily factorable. We look for two numbers that multiply to -2 and add to 1 (the coefficient of the x term). These numbers are 2 and -1. Thus, we can factor the quadratic equation as $(x + 2)(x - 1) = 0$. This factored form allows us to use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Applying this property, we set each factor equal to zero: $x + 2 = 0$ and $x - 1 = 0$. Solving these linear equations, we find the roots of the quadratic equation to be $x = -2$ and $x = 1$. These roots are the critical points that will help us determine the intervals where the original inequality $x^2 + x - 2 \geq 0$ holds true. These solutions are the lynchpin for our next step, where we analyze the intervals defined by these critical points.

Step 3: Determine the Intervals

The solutions to the quadratic equation, $x = -2$ and $x = 1$, serve as critical points that divide the number line into three distinct intervals. These intervals are: $(-\infty, -2)$, $(-2, 1)$, and $(1, \infty)$. Within each of these intervals, the quadratic expression $x^2 + x - 2$ will maintain a consistent sign—either positive or negative. This is because the expression can only change signs at the points where it equals zero, which are the critical points we found. To determine the solution set of the quadratic inequality $x^2 + x - 2 \geq 0$, we need to examine each interval and test a value within it to see if it satisfies the inequality. The intervals represent all possible values of x, and by analyzing them, we can identify the ranges where the quadratic expression is either greater than or equal to zero. This step is essential for piecing together the final solution set.

Step 4: Test Values in Each Interval

To determine the solution set of the quadratic inequality $x^2 + x - 2 \geq 0$, we must now test a value from each of the intervals we identified: $(-\infty, -2)$, $(-2, 1)$, and $(1, \infty)$. This process will reveal whether the quadratic expression is positive or negative within each interval. Let's choose test values for each interval:

1. For the interval $(-\infty, -2)$, we can choose $x = -3$. Substituting this into the inequality, we get $(-3)^2 + (-3) - 2 = 9 - 3 - 2 = 4$, which is greater than 0. Therefore, the inequality holds true in this interval.
2. For the interval $(-2, 1)$, we can choose $x = 0$. Substituting this into the inequality, we get $(0)^2 + (0) - 2 = -2$, which is less than 0. Therefore, the inequality does not hold true in this interval.
3. For the interval $(1, \infty)$, we can choose $x = 2$. Substituting this into the inequality, we get $(2)^2 + (2) - 2 = 4 + 2 - 2 = 4$, which is greater than 0. Therefore, the inequality holds true in this interval.

By testing these values, we can clearly see which intervals satisfy the original inequality. This testing process is a critical step in solving quadratic inequalities as it provides the empirical evidence needed to define the solution set.

Step 5: Include the Critical Points

In the previous step, we identified the intervals where the quadratic inequality $x^2 + x - 2 > 0$ holds true. However, the original inequality is $x^2 + x - 2 \geq 0$, which includes the condition where the expression equals zero. This means we must consider the critical points we found earlier, $x = -2$ and $x = 1$, as they are also part of the solution set. These points are where the quadratic expression equals zero, satisfying the "or equal to" part of the inequality. Therefore, we include these points in our solution set. The intervals where the inequality holds true are $(-\infty, -2)$ and $(1, \infty)$. By including the critical points, we are essentially closing these intervals at $x = -2$ and $x = 1$. This distinction is crucial because it accurately reflects the complete solution set of the inequality. Remembering to include these points is a key detail in solving quadratic inequalities correctly.

Step 6: Write the Solution Set

Now that we have identified the intervals where the quadratic inequality $x^2 + x - 2 \geq 0$ holds true, and we've included the critical points, we can write the final solution set. The inequality is satisfied for $x$ values in the intervals $(-\infty, -2]$ and $[1, \infty)$. This means that x is less than or equal to -2, or x is greater than or equal to 1. We can express this solution set using interval notation or set-builder notation. In set-builder notation, the solution set is written as {x | x ≤ -2 or x ≥ 1}. This notation clearly indicates that the solution set includes all x values that are either less than or equal to -2, or greater than or equal to 1. This is the comprehensive and accurate representation of the solution to the given quadratic inequality. Understanding how to write the solution set in the correct notation is the final step in mastering the solution of quadratic inequalities.

Conclusion

In conclusion, solving the quadratic inequality $x^2 + x - 2 \geq 0$ involves a series of methodical steps, from converting the inequality into an equation to writing the final solution set. We began by finding the critical points by solving the quadratic equation $x^2 + x - 2 = 0$, which gave us $x = -2$ and $x = 1$. These points divided the number line into intervals, which we then tested to determine where the inequality holds true. We found that the inequality is satisfied in the intervals $(-\infty, -2]$ and $[1, \infty)$. Finally, we expressed the solution set in set-builder notation as {x | x ≤ -2 or x ≥ 1}. This step-by-step process provides a clear framework for solving similar quadratic inequalities. Mastering these techniques is essential for anyone studying algebra and is a valuable skill in various mathematical and real-world applications. The ability to confidently solve quadratic inequalities is a testament to a solid understanding of algebraic principles.