solve of x: (x-3)(x+)>0
whoops sorry
(x-3)(x+3)>0
x can then be either >+3 or (both factors positive) or < -3 (both factors negative)
To solve the inequality (x-3)(x+)>0, we first need to find the values of x that satisfy it. Here's how you can proceed:
Step 1: Identify the critical points
To solve this inequality, we need to find the points where the expression (x-3)(x+) is equal to zero. The critical points occur when either (x-3)=0 or (x+)=0. Solving these equations, we find that x=3 and x=-.
Step 2: Analyze the intervals
Next, we need to understand the behavior of the inequality in different intervals. To do this, we can pick some test points within each interval to check the sign of the expression (x-3)(x+).
Interval 1: x<-
Pick a test point, let's say x=-1:
Substitute x=-1 into the expression: (-1-3)(-1+)=(-)(-)=1
Since 1 is greater than zero, the expression (x-3)(x+) is positive in this interval.
Interval 2: -<x<3
Pick a test point, let's say x=0:
Substitute x=0 into the expression: (0-3)(0+)=(-)(+)=-
Since - is negative, the expression (x-3)(x+) is negative in this interval.
Interval 3: 3<x<
Pick a test point, let's say x=4:
Substitute x=4 into the expression: (4-3)(4+)=(+)(+)=+
Since + is positive, the expression (x-3)(x+) is positive in this interval.
Step 3: Determine the solution
Based on the analysis of the intervals, we see that the expression (x-3)(x+) is positive in the intervals x<- and 3<x<. However, it is negative in the interval -<x<3.
To satisfy the inequality (x-3)(x+)>0, the expression must be positive. Therefore, the solution is the union of the intervals x<- and 3<x<. In interval notation, the solution is (-∞, -) ∪ (3, ∞).