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# Solving Inequalities Algebraically

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The sqrt(16) = 4, but the equation x2 = 16 has two solutions, x = 4, -4. The reason for this fact is the fact that sqrt(x2) = the absolute value of x. Using this fact, solve the following inequality algebraically:

x2-9 > 0

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Inequalities can be tricky things, so let's solve them. In this problem, we will solve for x in the equation $$x^2-9>0$$:

$$x^2-9>0$$ Add 9 to both sides of the inequality.
$$x^2>9$$ Take the square root of both sides.
$$|x|>\sqrt{9}$$
$$|x|>3$$ The absolute value splits the solutions into 2 inequalities. Solve them separately.
 $$x>3$$ $$-x>3$$

Divide by -1 on both sides. Remember that doing so flips the inequality sign. That's easy to forget!
 $$x>3$$ $$x<-3$$

Npw, let's combine this into a compound inequality, if possible. Unfortunately, in this case, it is not.

Therefore, x must either be greater than 3 or less than -3.

TheXSquaredFactor  Sep 9, 2017