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How many integers n satisfy \((n+3)(n-7) \le 0\)?

 Jul 26, 2019

Best Answer 

 #1
avatar+25228 
+3

How many integers n satisfy \(\large{(n+3)(n-7) \leq 0}\)?

 

We create a sign table:

\(\begin{array}{|l|c|c|c|c|} \hline \text{Interval or position} : & (-\infty,-3) & -3 & (-3,7) & 7 & (7,\infty) \\ \hline \text{sign of } (n+3): & - & 0 & + & + & + \\ \hline \text{sign of } (n-7): & - & - & - & 0 & + \\ \hline \text{sign of }(n+3)(n-7): & + & \color{red}0 & \color{red}- & \color{red}0 & + \\ \hline \end{array}\)

 

We can read the result: in the interval [-3,7] the left side of the inequality is negative or zero, and thus the inequality is true there.

 

\(-3\leq n \leq 7 \qquad n=\{-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7 \} \quad (11 \text{ integers, assumption zero is an integer }) \)

 

laugh

 Jul 26, 2019
 #1
avatar+25228 
+3
Best Answer

How many integers n satisfy \(\large{(n+3)(n-7) \leq 0}\)?

 

We create a sign table:

\(\begin{array}{|l|c|c|c|c|} \hline \text{Interval or position} : & (-\infty,-3) & -3 & (-3,7) & 7 & (7,\infty) \\ \hline \text{sign of } (n+3): & - & 0 & + & + & + \\ \hline \text{sign of } (n-7): & - & - & - & 0 & + \\ \hline \text{sign of }(n+3)(n-7): & + & \color{red}0 & \color{red}- & \color{red}0 & + \\ \hline \end{array}\)

 

We can read the result: in the interval [-3,7] the left side of the inequality is negative or zero, and thus the inequality is true there.

 

\(-3\leq n \leq 7 \qquad n=\{-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7 \} \quad (11 \text{ integers, assumption zero is an integer }) \)

 

laugh

heureka Jul 26, 2019
 #2
avatar+1198 
+1

Thanks heureka!!

 Jul 26, 2019
 #3
avatar+25228 
+1

Thank you, Logic !

 

laugh

heureka  Jul 28, 2019

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