How do you solve and graph the compound inequality #2 < x < 7# and #2x > 10# ?

#2x>10->x>5-> 5 < x# (because you may divide both sides by a positive number)

#5 < x<7#

To solve and graph the compound inequality \(2 < x < 7\) and \(2x > 10\), we need to find the values of \(x\) that satisfy both conditions. For the first inequality \(2 < x < 7\), \(x\) must be greater than 2 and less than 7. For the second inequality \(2x > 10\), we solve for \(x\): \[2x > 10\] Divide both sides by 2: \[x > 5\] So, the values of \(x\) satisfying the second inequality are greater than 5. To find the intersection of both inequalities, we consider the values that satisfy both conditions: \[2 < x < 7\] and \(x > 5\] Combining these conditions, we find that \(x\) must be greater than 5 and less than 7. Therefore, the solution to the compound inequality is \(5 < x < 7\). To graph this compound inequality on a number line, we mark an open circle at 5 (since \(x\) cannot be equal to 5) and at 7 (since \(x\) cannot be equal to 7), and then shade the region between these two points. This represents all values of \(x\) greater than 5 and less than 7. The graph should show an open circle at 5, an open circle at 7, and a shaded region between them.