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Chapter 2: Problem 17

Solve the inequality. Then graph the solution set. $$x^{2}+4 x+4 \geq 9$$

Short Answer

Expert verified
The solution of the inequality \(x^{2}+4 x+4 \geq 9\) is the closed interval [-5, 1]. This is represented on a number line with closed circles at -5 and 1, and a line segment that connects these points.

Step by step solution

01

Subtract 9 from both sides

Subtract 9 from both sides of the inequality to set it to zero, giving: \( x^{2} + 4x + 4 - 9 \geq 0 \), which simplifies to: \( x^{2} + 4x - 5 \geq 0 \)
02

Factoring the expression

Now, let's factor the quadratic expression on the left side of the equation. It can be factored into: \( (x - 1)(x + 5) \geq 0 \)
03

Solution Set

Assign the factors equal to zero to find their respective 'x' values. We get x = 1 and x = -5.
04

Identify intervals and test inequality

The values -5, 1 divide the number line into three intervals: (-inf,-5), (-5,1) and (1,inf). We need to select a test point from each interval and substitute it back into our inequality. If the inequality holds true, all numbers in that interval are a part of the solution. On doing it: Interval 1: Choose x = -6, into \( (x - 1)(x + 5) \geq 0 \), It is False. Interval 2: Choose x = 0, It is True. Interval 3: Choose x = 2, It is False. So the solution interval is [-5,1].
05

Plot on graph

Plot the solution interval [-5,1] on the number line. Closed circles at -5 and 1 with a line segment connecting them represent the solution.

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