Solve the following inequality. $$ x^{2}-13 x+220 $$

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Solve the following inequality. $$ x^{2}-13 x+22>0 $$

UpStudy AI · Accepted Answer

1. Rewrite the inequality as $$ x^2 - 13x + 22 > 0. $$ 2. Factor the quadratic expression. We look for two numbers that multiply to $$22$$ and add to $$-13$$. These numbers are $$-2$$ and $$-11$$, so we have $$ x^2 - 13x + 22 = (x - 2)(x - 11). $$ 3. The inequality becomes $$ (x - 2)(x - 11) > 0. $$ 4. Identify the critical points by setting each factor equal to zero: $$ x - 2 = 0 \quad \Rightarrow \quad x = 2, $$ $$ x - 11 = 0 \quad \Rightarrow \quad x = 11. $$ 5. Analyze the intervals determined by the critical points: - For $$x < 2$$: Choose $$x = 0$$. Then, $$ (0-2)(0-11) = (-2)(-11) = 22 > 0. $$ - For $$2 < x < 11$$: Choose $$x = 7$$. Then, $$ (7-2)(7-11) = (5)(-4) = -20 < 0. $$ - For $$x > 11$$: Choose $$x = 12$$. Then, $$ (12-2)(12-11) = (10)(1) = 10 > 0. $$ 6. The inequality holds for the intervals where the product is positive: $$ x \in (-\infty, 2) \quad \text{or} \quad x \in (11, \infty). $$ Thus, the solution is $$ (-\infty, 2) \cup (11, \infty). $$ The solution to the inequality $$ x^{2} - 13x + 22 > 0 $$ is all real numbers less than 2 or greater than 11.

Solve the following inequality.

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