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

Jul 26, 2019

#1
+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.

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

Jul 26, 2019

#1
+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.

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

heureka Jul 26, 2019
#2
+1198
+1

Thanks heureka!!

Jul 26, 2019
#3
+25228
+1

Thank you, Logic !

heureka  Jul 28, 2019