Solving $x^2 + X - 12 \leqslant 0$: A Step-by-Step Guide

Let's dive into solving the inequality $x^2 + x - 12 \leqslant 0$. Inequalities like these pop up frequently in mathematics, especially in algebra and calculus, so mastering them is super important. We’ll break down the process step by step, making it easy to understand and apply. This guide aims to provide a clear and comprehensive approach to solving quadratic inequalities, ensuring you grasp not just the how, but also the why behind each step.

1. Factoring the Quadratic Expression

The first step in solving any quadratic inequality is to deal with the quadratic expression itself. In our case, we have $x^2 + x - 12$. The goal here is to factor this expression into two binomials. Factoring simplifies the inequality, making it easier to find the critical points. To factor $x^2 + x - 12$, we need to find two numbers that multiply to -12 and add up to 1 (the coefficient of the x term). Think of the pairs of factors for -12: (-1, 12), (-2, 6), (-3, 4), (1, -12), (2, -6), and (3, -4). Among these, -3 and 4 satisfy our condition because (-3) * 4 = -12 and (-3) + 4 = 1. Therefore, we can rewrite the quadratic expression as $(x - 3)(x + 4)$. This factorization is a crucial step because it transforms the quadratic inequality into a product of two linear factors, which are much simpler to analyze. Understanding factorization not only helps in solving inequalities but also in various other areas of mathematics, such as finding roots of quadratic equations and simplifying algebraic expressions. This initial step sets the foundation for the subsequent steps, making the entire process more manageable. Remember, the accuracy of your factorization directly impacts the correctness of your final solution, so take your time and double-check your work.

2. Identifying the Critical Points

After factoring the quadratic expression, we have $(x - 3)(x + 4) leq 0$. Now, we need to find the critical points. Critical points are the values of $x$ that make the expression equal to zero. These points are crucial because they divide the number line into intervals, within which the expression will have a consistent sign (either positive or negative). To find these critical points, we set each factor equal to zero: $x - 3 = 0$ and $x + 4 = 0$. Solving these equations gives us $x = 3$ and $x = -4$. These are our critical points. They are the points where the quadratic expression changes its sign. The importance of identifying critical points cannot be overstated. They serve as boundaries that define the intervals where the inequality's solution lies. By finding these points, we are essentially pinpointing where the graph of the quadratic function intersects the x-axis, thereby changing from positive to negative or vice versa. This step is not just about finding numbers; it's about understanding the behavior of the quadratic expression. Accurately determining the critical points is vital for constructing the sign chart in the next step, which is instrumental in identifying the solution intervals. Therefore, careful attention to detail in this step will pave the way for a correct and complete solution.

3. Creating a Sign Chart

With the critical points identified, we create a sign chart. A sign chart is a visual tool that helps us determine the sign of the expression $(x - 3)(x + 4)$ in the intervals defined by the critical points. Draw a number line and mark the critical points, -4 and 3. These points divide the number line into three intervals: $(-\infty, -4)$, $(-4, 3)$, and $(3, \infty)$. Now, we need to test a value from each interval in the factored expression $(x - 3)(x + 4)$ to determine its sign in that interval. For the interval $(-\infty, -4)$, let's test $x = -5$. Plugging this into our expression, we get $(-5 - 3)(-5 + 4) = (-8)(-1) = 8$, which is positive. For the interval $(-4, 3)$, let's test $x = 0$. We get $(0 - 3)(0 + 4) = (-3)(4) = -12$, which is negative. For the interval $(3, \infty)$, let's test $x = 4$. We get $(4 - 3)(4 + 4) = (1)(8) = 8$, which is positive. We can now represent this information on the sign chart:

        (-\infty, -4)   |   (-4, 3)   |   (3, \infty)
--------------------|------------|------------------
(x - 3)      -          |     -      |        +
(x + 4)      -          |     +      |        +
(x - 3)(x + 4)  +          |     -      |        +

The sign chart is a powerful visual aid that translates algebraic expressions into a clear picture of their behavior across different intervals. It helps to avoid mistakes and provides a structured approach to solving inequalities. The key to using a sign chart effectively is to choose test values that are easy to work with and representative of their respective intervals. This systematic approach ensures that we accurately capture the sign of the expression in each interval, which is crucial for identifying the solution set. The sign chart not only simplifies the process but also deepens the understanding of how the factors of the quadratic expression interact to determine the overall sign.

4. Determining the Solution Set

Looking at the sign chart, we want to find the intervals where $(x - 3)(x + 4) leq 0$. This means we are looking for intervals where the expression is either negative or zero. From the sign chart, we see that the expression is negative in the interval $(-4, 3)$. Additionally, we need to include the critical points because the inequality is non-strict ($\leqslant$), meaning the expression can be equal to zero. The critical points are $x = -4$ and $x = 3$. Therefore, the solution set includes these points. Combining the interval where the expression is negative and the critical points, we get the solution set $[-4, 3]$. This interval notation means that the solution includes all values of $x$ greater than or equal to -4 and less than or equal to 3. Understanding how to interpret the sign chart and translate it into a solution set is a critical skill in solving inequalities. The solution set represents all values of $x$ that satisfy the original inequality, and accurately identifying this set is the ultimate goal. The process involves not only reading the signs from the chart but also considering whether the inequality is strict (< or >) or non-strict (< or >), as this determines whether the critical points should be included in the solution. This final step ties together all the previous steps, culminating in a clear and concise answer to the problem.

5. Writing the Solution

Finally, we write the solution in the appropriate notation. The solution to the inequality $x^2 + x - 12 leq 0$ is $x ∈ [-4, 3]$. This interval notation indicates that the solution includes all real numbers $x$ between -4 and 3, including -4 and 3 themselves. Alternatively, we can write the solution as $-4 leq x leq 3$, which conveys the same information in inequality notation. The importance of expressing the solution clearly and correctly cannot be overstated. The solution set is the final answer, and it should be presented in a manner that is both accurate and easily understood. Whether using interval notation or inequality notation, the key is to ensure that the solution accurately reflects the values of $x$ that satisfy the original inequality. This final step provides closure to the problem-solving process, demonstrating a comprehensive understanding of how to solve quadratic inequalities. Mastering this step allows you to confidently communicate your solutions in a clear and professional manner, which is essential in mathematical contexts.

In conclusion, solving the inequality $x^2 + x - 12 leq 0$ involves factoring the quadratic expression, identifying critical points, creating a sign chart, determining the solution set, and writing the solution in the appropriate notation. Each step is crucial and builds upon the previous one. By following these steps, you can confidently solve similar quadratic inequalities. Remember, practice makes perfect, so try solving various quadratic inequalities to solidify your understanding. Understanding these concepts is not just about getting the right answer; it’s about developing a deeper appreciation for mathematical problem-solving and analytical thinking. If you're interested in learning more about inequalities and related topics, you might find valuable resources on websites like Khan Academy. Their explanations and practice exercises can further enhance your understanding.