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Chapter 19: Problem 582

Find the solution set of \(-x^{2}-4 x-5>0\)

Short Answer

Expert verified
The solution set for the given inequality \(-x^2 - 4x - 5 > 0\) is the empty set, as there are no real values of \(x\) for which the inequality is true. This is because the quadratic equation \(-x^2 - 4x - 5 = 0\) has no real solutions, and after testing a point within the domain, we determined that the inequality doesn't hold true for any real value of \(x\).

Step by step solution

01

Solve the quadratic equation

First, solve the equation \(-x^2 - 4x - 5 = 0\). To do this, you can use either the quadratic formula or factoring. Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we get: \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-1)(-5)}}{2(-1)}\) Simplifying, we have: \(x = \frac{4 \pm \sqrt{16 - 20}}{-2}\) \(x = \frac{4 \pm \sqrt{-4}}{-2}\) We can see that there is no real solution for the equation, as the square root of a negative number is imaginary. It means that the inequality doesn't change the sign.
02

Test the intervals

Since there are no boundary points, we can choose any point in the domain to test if the inequality holds true for that value of \(x\). Let's test the point \(x = 0\): \(-0^2 - 4(0) - 5 > 0\) Simplifying, we get: \(-5 > 0\) This is clearly false, so the inequality \(-x^2 - 4x - 5 > 0\) doesn't hold true for any real value of \(x\). Therefore, the solution set for the given inequality is the empty set, meaning there are no real values of \(x\) for which the inequality is true.

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Most popular questions from this chapter

Sketch the graph of the function \(\left\\{\left(\mathrm{x}, \mathrm{x}^{2}\right)\right\\}\).

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Solve: \(\mathrm{x}^{2}<2 \mathrm{x}+2\)

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Draw a graph of the set of ordered pairs which satisfy the function \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-7\)
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