Solve and graph the compound inequality on a number line.
x/4 < −1 and x + 6 > −2
To solve the compound inequality x/4 < -1 and x + 6 > -2, start by solving each inequality separately.
For the first inequality, x/4 < -1, multiply both sides of the inequality by 4 to isolate x:
x < -4
For the second inequality, x + 6 > -2, subtract 6 from both sides to isolate x:
x > -8
Now, we have x < -4 and x > -8. To graph these inequalities on a number line, first mark -4 and -8 on the number line. Then, shade the region to the left of -4 (indicating x < -4) and the region to the right of -8 (indicating x > -8). The shaded regions should overlap from -4 to -8.
Here is how the graph would look:
\[
\begin{array}{cccccccccc}
& & & & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\
& & & & -10 & -9 & -8 & -7 & -6 & -5 & -4 \\
\end{array}
\]
Note: The filled circles indicate that -4 and -8 are not included in the solution set, since the inequalities are strict (x < -4 and x > -8).