How do you solve the inequality: #(x - 3) (x - 5) > 0#?

In order for this inequality to be true, you need #(x-3)# and #(x-5)# to either both be positive or both be negative.

For any value of #x>5# you will get

#{(x-3 > 0), (x-5>0) :} implies (x-3)(x-5)>0#

For any value of #x<3# you will get

#{(x-3<0), (x-5<0):} implies (x-3)(x-5)>0#

This inequality will thus be satisfied for any value of #x in (-oo, 3) uu (5, +oo)#.

On the other hand, any value of #x in [3, 5]# will not be a valid solution.

To solve the inequality (x - 3)(x - 5) > 0: 1. Find the critical points by setting each factor equal to zero and solving for x: x - 3 = 0 --> x = 3 x - 5 = 0 --> x = 5 2. Plot these critical points on a number line. 3. Test intervals between the critical points by choosing test points within each interval and determining whether the expression is positive or negative in each interval. 4. Analyze the signs of the expression in each interval to determine the solution set. - If (x - 3)(x - 5) > 0, then either both factors are positive or both factors are negative. - For x < 3, both factors are negative, so the expression is positive. - For 3 < x < 5, one factor is positive and the other is negative, so the expression is negative. - For x > 5, both factors are positive, so the expression is positive. 5. Therefore, the solution to the inequality is: x < 3 or x > 5.