Graphing The Solution To (x+4)(x-7) ≤ 0 On A Number Line

In this article, we will delve into the process of graphing the solution to the inequality $(x+4)(x-7) \,\leq 0$ on the number line. This type of problem is a fundamental concept in algebra, particularly when dealing with quadratic inequalities. Understanding how to solve and represent these inequalities graphically is crucial for various mathematical applications. We will break down the steps involved, providing a clear, comprehensive guide suitable for students and anyone looking to refresh their algebra skills. This exploration will cover identifying critical points, testing intervals, and accurately plotting the solution set on the number line, ensuring a solid grasp of the concepts involved.

To effectively graph the solution, it's essential to first understand the inequality $(x+4)(x-7) \leq 0$. This is a quadratic inequality, which means we are looking for the values of $x$ that make the expression $(x+4)(x-7)$ less than or equal to zero. The expression is a product of two linear factors: $(x+4)$ and $(x-7)$. The inequality holds true when this product is either negative or zero. To find the intervals where this occurs, we need to identify the critical points, which are the values of $x$ that make the expression equal to zero. These points serve as boundaries that divide the number line into intervals that we need to test. The critical points are crucial because they represent the points where the expression changes sign. This is because a continuous function (like a polynomial) can only change its sign at its zeros. Thus, by finding these zeros, we can divide the number line into intervals and test a value within each interval to determine whether the expression is positive or negative in that interval. This method allows us to systematically solve the inequality and identify the solution set.

The first step in graphing the solution is to find the critical points of the inequality. Critical points are the values of $x$ that make the expression $(x+4)(x-7)$ equal to zero. To find these points, we set each factor equal to zero and solve for $x$:

1. $x + 4 = 0$
   Subtracting 4 from both sides gives $x = -4$.
2. $x - 7 = 0$
   Adding 7 to both sides gives $x = 7$.

Thus, the critical points are $x = -4$ and $x = 7$. These critical points are crucial because they divide the number line into three intervals: $(-\infty, -4)$, $(-4, 7)$, and $(7, \infty)$. Each of these intervals needs to be tested to determine whether the inequality $(x+4)(x-7) \leq 0$ holds true. The critical points themselves are also important because the inequality includes "equal to zero," so these points will be part of the solution set. They act as the boundaries that define the intervals where the inequality is either true or false. Therefore, accurately identifying these critical points is the foundation for graphing the solution set on the number line.

Once we have identified the critical points, the next step is to test the intervals created by these points. The critical points $x = -4$ and $x = 7$ divide the number line into three intervals: $(-\infty, -4)$, $(-4, 7)$, and $(7, \infty)$. To determine whether the inequality $(x+4)(x-7) \leq 0$ holds true in each interval, we select a test value within each interval and substitute it into the inequality.

1. For the interval $(-\infty, -4)$, let's choose $x = -5$. Substituting this value into the inequality gives us:

   $$((-5) + 4)((-5) - 7) = (-1)(-12) = 12$$

   Since $12 > 0$, the inequality does not hold true in this interval.

2. For the interval $(-4, 7)$, let's choose $x = 0$. Substituting this value into the inequality gives us:

   $$(0 + 4)(0 - 7) = (4)(-7) = -28$$

   Since $-28 \leq 0$, the inequality holds true in this interval.

3. For the interval $(7, \infty)$, let's choose $x = 8$. Substituting this value into the inequality gives us:

   $$(8 + 4)(8 - 7) = (12)(1) = 12$$

   Since $12 > 0$, the inequality does not hold true in this interval.

By testing these intervals, we can determine which sections of the number line satisfy the given inequality. This process is crucial for accurately graphing the solution set.

Now that we have identified the critical points and tested the intervals, we can graph the solution to the inequality $(x+4)(x-7) \leq 0$ on the number line. The critical points are $x = -4$ and $x = 7$. We found that the inequality holds true in the interval $(-4, 7)$. Additionally, since the inequality includes "less than or equal to," we also include the critical points themselves in the solution. This means that $x = -4$ and $x = 7$ are part of the solution set.

To represent this graphically, we draw a number line and mark the critical points $x = -4$ and $x = 7$. Since these points are included in the solution, we use closed circles (or filled-in dots) at these points. For the interval $(-4, 7)$, where the inequality holds true, we draw a line segment connecting the two closed circles. This line segment represents all the values of $x$ between $-4$ and $7$, inclusive, that satisfy the inequality. The intervals $(-\infty, -4)$ and $(7, \infty)$ do not satisfy the inequality, so we do not include them in the graph.

The final graph will show a line segment between $-4$ and $7$, with closed circles at both endpoints. This visual representation clearly indicates the solution set of the inequality $(x+4)(x-7) \leq 0$. The graph serves as a comprehensive tool for understanding the range of $x$ values that make the inequality true, making it easier to interpret and communicate the solution.

In conclusion, graphing the solution to the inequality $(x+4)(x-7) \leq 0$ on the number line involves several key steps: finding the critical points, testing the intervals, and representing the solution graphically. By setting each factor equal to zero, we identified the critical points $x = -4$ and $x = 7$. These points divide the number line into intervals that we tested to determine where the inequality holds true. We found that the inequality is satisfied in the interval $(-4, 7)$, and because the inequality includes "equal to zero," the critical points themselves are also part of the solution. Graphically, this is represented by a line segment connecting $-4$ and $7$ on the number line, with closed circles at both endpoints to indicate that these points are included. This process is not only applicable to this specific problem but can be generalized to solve other quadratic inequalities as well.

Mastering these steps provides a strong foundation for solving more complex mathematical problems involving inequalities. The ability to accurately identify critical points, test intervals, and graphically represent solutions is an invaluable skill in algebra and beyond. By understanding these fundamental concepts, students and practitioners alike can confidently tackle a wide range of mathematical challenges. The methods and principles discussed in this article serve as a comprehensive guide for anyone looking to deepen their understanding of quadratic inequalities and their graphical solutions.