Solve: When Is -x² - 4x + 5 > -2x - 3?

Hey guys! Let's dive into a classic math problem where we need to figure out when a quadratic graph sits higher than a linear graph. Specifically, we want to find the values of x for which the graph of the quadratic equation y = -x² - 4x + 5 is above the graph of the linear equation y = -2x - 3. This means we need to determine when the y-values of the quadratic equation are greater than the y-values of the linear equation. So, let's break it down step-by-step.

Setting up the Inequality

First things first, we need to express our problem mathematically. We want to find when the quadratic expression is greater than the linear expression. This gives us the inequality:

-x² - 4x + 5 > -2x - 3

This inequality is the key to solving our problem. It tells us exactly when the quadratic function’s output (y-value) is larger than the linear function’s output. To solve this inequality, we need to rearrange it into a more manageable form. Our goal is to get all the terms on one side, leaving zero on the other side. This will allow us to analyze the quadratic expression more easily. Think of it like preparing the equation for factoring or using the quadratic formula – we need it in a standard form to proceed effectively. By moving all terms to one side, we can clearly see the quadratic expression we're dealing with and identify its coefficients, which are crucial for further steps. This sets us up for either factoring the quadratic, completing the square, or using the quadratic formula, all of which require the quadratic to be in the standard form of ax² + bx + c.

Rearranging the Inequality

To rearrange, let’s add 2x and 3 to both sides of the inequality. This will help us consolidate the terms and simplify the expression:

-x² - 4x + 5 + 2x + 3 > 0

Now, we combine like terms:

-x² - 2x + 8 > 0

To make it easier to work with, let's multiply the entire inequality by -1. Remember, when we multiply or divide an inequality by a negative number, we need to flip the inequality sign. So, we get:

x² + 2x - 8 < 0

Now we have a quadratic inequality in a standard form that is easier to solve. The act of multiplying by -1 and flipping the inequality sign is a crucial step. It transforms the inequality into a form where the leading coefficient (the coefficient of x²) is positive, which makes the subsequent steps of factoring or using the quadratic formula much simpler. If we had left the inequality with a negative leading coefficient, we would still be able to solve it, but it adds an extra layer of complexity and potential for error. This step is all about simplifying the algebra to make the problem more approachable.

Factoring the Quadratic

Now, let’s factor the quadratic expression x² + 2x - 8. We're looking for two numbers that multiply to -8 and add up to 2. Those numbers are 4 and -2. So, we can factor the quadratic as follows:

(x + 4)(x - 2) < 0

Factoring the quadratic expression is a pivotal step because it allows us to identify the critical points where the expression changes sign. These critical points are the roots of the quadratic equation, which are the x-values that make the expression equal to zero. In our case, these are the values of x that make either (x + 4) or (x - 2) equal to zero. By finding these points, we divide the number line into intervals, and within each interval, the sign of the quadratic expression remains consistent (either positive or negative). This makes it much easier to determine the range of x-values for which the inequality holds true. The ability to factor quadratics efficiently is a fundamental skill in algebra, and it's particularly useful for solving inequalities and understanding the behavior of quadratic functions.

Finding the Critical Points

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

x + 4 = 0 => x = -4

x - 2 = 0 => x = 2

These critical points, x = -4 and x = 2, are where the quadratic expression equals zero. They divide the number line into three intervals: x < -4, -4 < x < 2, and x > 2. These critical points are incredibly important because they mark the boundaries where the quadratic expression can change its sign. Between these points, the expression will either be consistently positive or consistently negative. This is because a quadratic function, represented by a parabola, can only change its sign at its roots (the points where it intersects the x-axis). By identifying these critical points, we can systematically test each interval to determine where the inequality is satisfied. This method of using critical points is a fundamental technique for solving inequalities, not just for quadratics, but also for higher-degree polynomials and rational functions. It provides a structured approach to breaking down a complex problem into manageable parts.

Testing Intervals

Now, we need to test a value from each interval in the inequality (x + 4)(x - 2) < 0 to see where the expression is negative:

1. Interval 1: x < -4

   Let’s pick x = -5:

   (-5 + 4)(-5 - 2) = (-1)(-7) = 7 > 0 (Positive)

   So, the inequality is not satisfied in this interval.

2. Interval 2: -4 < x < 2

   Let’s pick x = 0:

   (0 + 4)(0 - 2) = (4)(-2) = -8 < 0 (Negative)

   So, the inequality is satisfied in this interval.

3. Interval 3: x > 2

   Let’s pick x = 3:

   (3 + 4)(3 - 2) = (7)(1) = 7 > 0 (Positive)

   So, the inequality is not satisfied in this interval.

Testing intervals is a crucial step in solving inequalities because it allows us to determine the sign of the expression in each region defined by the critical points. By choosing a test value within each interval and plugging it into the factored inequality, we can quickly see whether the expression is positive or negative. This method works because the sign of the expression can only change at the critical points. If the expression is negative for one value in an interval, it will be negative for all values in that interval. This systematic approach ensures that we don't have to test every single value of x; instead, we only need one representative from each interval. This significantly simplifies the process of finding the solution set for the inequality.

The Solution

The inequality (x + 4)(x - 2) < 0 is satisfied when -4 < x < 2. This means that the quadratic graph is above the linear graph for x values between -4 and 2. Thus, the correct answer is:

C. -4 < x < 2

Wrapping it up, solving this type of problem involves several key steps: setting up the inequality, rearranging it into a standard form, factoring the quadratic expression, finding the critical points, testing intervals, and finally, determining the solution. Each step is important and builds upon the previous one to lead us to the correct answer. Remember, guys, practice makes perfect, so keep at it, and you'll become a pro at solving these types of problems!