Explanation

To solve the inequality x−3/x+7 < 0, we need to find where the expression is negative.

We first determine the critical points by setting the numerator and denominator equal to zero:

x−3 = 0 gives x = 3, and x+7 = 0 gives x = −7.

Now, we create a sign chart with these critical points and test intervals to see where the expression is negative:

Test x = −8 (a number less than -7): −8−3/-8+7 = −11/-1 ​= 11>0 (positive)

Test x = 0 (a number between -7 and 3): 0−3/0+7 = −3/7 <0 (negative)

Test x = 4 (a number greater than 3): 4−3/ 4+7 = 1/11 >0 (positive)

From this, we see the expression is negative when x is between -7 and 3, so the solution is (−7,3).