Solving Polynomial Inequality X^2 + 5x - 36 ≥ 0 A Step By Step Guide

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Polynomial inequalities, like the one presented, x^2 + 5x - 36 ≥ 0, are a fundamental topic in algebra and are essential for various mathematical and scientific applications. Mastering the techniques to solve these inequalities is crucial for students and professionals alike. This article provides a detailed, step-by-step guide on how to solve the polynomial inequality x^2 + 5x - 36 ≥ 0, ensuring a clear understanding of the underlying concepts and methods. We will break down the process into manageable steps, from factoring the quadratic expression to interpreting the solution set, making it accessible for learners of all levels.

Understanding Polynomial Inequalities

Before diving into the solution, let's first understand what polynomial inequalities are and why they are important. A polynomial inequality is a mathematical statement that compares a polynomial expression to a constant, typically zero, using inequality symbols such as >, <, ≥, or ≤. These inequalities arise in various contexts, including optimization problems, calculus, and real-world applications where quantities have constraints or limitations.

Solving a polynomial inequality involves finding the range(s) of values for the variable (in this case, x) that satisfy the inequality. This often requires algebraic manipulation, factoring, and analysis of the polynomial's behavior. The solution set represents the interval(s) on the number line where the polynomial expression meets the specified inequality condition. For example, in our case, we need to find the values of x for which the quadratic expression x^2 + 5x - 36 is greater than or equal to zero.

Polynomial inequalities are not just theoretical exercises; they have practical applications in numerous fields. In economics, they can be used to model constraints on production costs or revenue. In physics, they can describe the range of possible values for physical quantities like velocity or acceleration. In computer science, they appear in optimization algorithms and resource allocation problems. Therefore, a solid grasp of solving polynomial inequalities is an invaluable skill.

Step 1: Factoring the Quadratic Expression

The first critical step in solving the inequality x^2 + 5x - 36 ≥ 0 is to factor the quadratic expression on the left-hand side. Factoring transforms the quadratic into a product of two linear expressions, which makes it easier to analyze its sign. In other words, we are looking for two binomials that, when multiplied together, give us x^2 + 5x - 36.

To factor a quadratic expression of the form ax^2 + bx + c, we look for two numbers that multiply to ac and add up to b. In our case, a = 1, b = 5, and c = -36. Therefore, we need two numbers that multiply to -36 and add up to 5. By considering the factors of 36, we can quickly identify that the numbers 9 and -4 satisfy these conditions because 9 * (-4) = -36 and 9 + (-4) = 5.

Now, we can rewrite the middle term of the quadratic expression using these two numbers:

x^2 + 5x - 36 = x^2 + 9x - 4x - 36

Next, we factor by grouping. We group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group:

x^2 + 9x - 4x - 36 = x(x + 9) - 4(x + 9)

Notice that both terms now have a common factor of (x + 9). We can factor this out:

x(x + 9) - 4(x + 9) = (x - 4)(x + 9)

So, the factored form of the quadratic expression x^2 + 5x - 36 is (x - 4)(x + 9). This factorization is a crucial step because it allows us to identify the roots of the corresponding quadratic equation, which are essential for solving the inequality.

Step 2: Finding the Critical Points

Once we have factored the quadratic expression, the next step is to find the critical points or zeros of the expression. These are the values of x for which the expression equals zero. In the context of inequalities, critical points are the boundary values that divide the number line into intervals, within which the expression has a consistent sign (either positive or negative).

To find the critical points, we set each factor equal to zero and solve for x:

  1. x - 4 = 0 Adding 4 to both sides, we get: x = 4

  2. x + 9 = 0 Subtracting 9 from both sides, we get: x = -9

So, the critical points for the inequality x^2 + 5x - 36 ≥ 0 are x = 4 and x = -9. These two points are significant because they are the points where the quadratic expression changes its sign. On either side of these points, the expression will be either positive or negative. This is a direct consequence of the factored form of the quadratic and the fact that the sign of a product depends on the signs of its factors.

The critical points divide the number line into three intervals: (-∞, -9), (-9, 4), and (4, ∞). We need to determine the sign of the quadratic expression within each of these intervals. This can be done by choosing a test value from each interval and substituting it into the factored form of the expression. The sign of the result will tell us the sign of the expression in that entire interval.

Step 3: Testing Intervals

Now that we have identified the critical points x = -9 and x = 4, we need to determine the sign of the quadratic expression (x - 4)(x + 9) in each of the three intervals they define: (-∞, -9), (-9, 4), and (4, ∞). To do this, we will choose a test value within each interval and substitute it into the factored form of the expression. The sign of the result will indicate the sign of the expression throughout the interval.

  1. Interval (-∞, -9): Let's choose a test value less than -9, say x = -10. Substituting x = -10 into the factored expression: (-10 - 4)(-10 + 9) = (-14)(-1) = 14 The result is positive (14 > 0), so the expression is positive in the interval (-∞, -9).

  2. Interval (-9, 4): Let's choose a test value between -9 and 4, say x = 0. Substituting x = 0 into the factored expression: (0 - 4)(0 + 9) = (-4)(9) = -36 The result is negative (-36 < 0), so the expression is negative in the interval (-9, 4).

  3. Interval (4, ∞): Let's choose a test value greater than 4, say x = 5. Substituting x = 5 into the factored expression: (5 - 4)(5 + 9) = (1)(14) = 14 The result is positive (14 > 0), so the expression is positive in the interval (4, ∞).

By testing these intervals, we have determined the sign of the quadratic expression in each region defined by the critical points. This information is crucial for determining the solution set of the original inequality.

Step 4: Determining the Solution Set

With the sign of the expression determined in each interval, we can now find the solution set for the inequality x^2 + 5x - 36 ≥ 0. The inequality asks for values of x for which the expression is greater than or equal to zero. This means we are looking for the intervals where the expression is positive or zero.

From our interval testing, we found that the expression is positive in the intervals (-∞, -9) and (4, ∞). Additionally, the expression is equal to zero at the critical points x = -9 and x = 4. Since the inequality includes