QuestionJuly 20, 2025

Solve the inequality: n^2gt 7n+30 Give your answer in interval notation. Enter DNE if there is no solution. square

Solve the inequality: n^2gt 7n+30 Give your answer in interval notation. Enter DNE if there is no solution. square
Solve the inequality: n^2gt 7n+30
Give your answer in interval notation. Enter DNE if there is no solution.
square

Solution
4.4(259 votes)

Answer

(-\infty, -3) \cup (10, \infty) Explanation 1. Rearrange the inequality Move all terms to one side: n^2 - 7n - 30 > 0. 2. Factor the quadratic expression Factor as (n - 10)(n + 3) > 0. 3. Determine critical points Set each factor to zero: n - 10 = 0 \Rightarrow n = 10; n + 3 = 0 \Rightarrow n = -3. 4. Test intervals around critical points Test intervals: (-\infty, -3), (-3, 10), (10, \infty). - For n 0 \Rightarrow (negative)(negative) > 0 is true. - For -3 0 \Rightarrow (negative)(positive) > 0 is false. - For n > 10, choose n = 11: (11 - 10)(11 + 3) > 0 \Rightarrow (positive)(positive) > 0 is true. 5. Write solution in interval notation Combine true intervals: (-\infty, -3) \cup (10, \infty).

Explanation

1. Rearrange the inequality<br /> Move all terms to one side: $n^2 - 7n - 30 > 0$.<br />2. Factor the quadratic expression<br /> Factor as $(n - 10)(n + 3) > 0$.<br />3. Determine critical points<br /> Set each factor to zero: $n - 10 = 0 \Rightarrow n = 10$; $n + 3 = 0 \Rightarrow n = -3$.<br />4. Test intervals around critical points<br /> Test intervals: $(-\infty, -3)$, $(-3, 10)$, $(10, \infty)$.<br />- For $n < -3$, choose $n = -4$: $(-4 - 10)(-4 + 3) > 0 \Rightarrow (negative)(negative) > 0$ is true.<br />- For $-3 < n < 10$, choose $n = 0$: $(0 - 10)(0 + 3) > 0 \Rightarrow (negative)(positive) > 0$ is false.<br />- For $n > 10$, choose $n = 11$: $(11 - 10)(11 + 3) > 0 \Rightarrow (positive)(positive) > 0$ is true.<br />5. Write solution in interval notation<br /> Combine true intervals: $(-\infty, -3) \cup (10, \infty)$.
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