+26367 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.
\(-3\leq n \leq 7 \qquad n=\{-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7 \} \quad (11 \text{ integers, assumption zero is an integer }) \)
+26367 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.
\(-3\leq n \leq 7 \qquad n=\{-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7 \} \quad (11 \text{ integers, assumption zero is an integer }) \)
