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Chapter 2: Problem 22

Solve the inequality. Then graph the solution set. $$x^{2}>2 x+8$$

Short Answer

Expert verified
The solution to the inequality \(x^{2}-2x-8>0\) is that \(x\) is greater than 4 or \(x\) is less than -2, denoted in interval notation as (-infinity,-2)U(4,infinity). This corresponds to a graph with an open circle at -2 and 4, and shading in between these points.

Step by step solution

01

Rearrange the inequality

The first step is to arrange the inequality in standard form. This means bringing all terms to one side. So, the inequality \(x^{2}>2 x+8\) is rearranged to \(x^{2}-2x-8 > 0\)
02

Factor the quadratic

Next, the quadratic needs to be factored. This will give us the critical points where the function intersects the x-axis. Factoring the equation \(x^{2}-2x-8\) results in \((x-4)(x+2) > 0\)
03

Finding the critical points

The roots of the equation, also known as the critical points, are found by setting the equation equal to zero and solving for x. This gives us \(x-4 = 0\) (or \(x = 4\)) and \(x+2 = 0\) (or \(x = -2\)). These are the points where the function intersects the x-axis
04

Determine solution intervals

Now, look at the intervals determined by the critical points: (-infinity, -2), (-2, 4), and (4, infinity). Test each interval by plugging in a test point into the factored inequality. If the inequality is true, then that interval is part of the solution set; if it's false, then it is not.
05

Graph the solution set

Lastly, graph the solution set. Because the question asked for a greater than inequality, rather than greater than or equal to, we use an open circle at -2 and 4. Our solution from the previous step tells us to shade in between -2 and 4, and not any part outside those points.

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