How to solve this:
((x-3)(x+2))/((x-1)) < 0
I don't understand since theres x-1 in the denominator
The critical values are -2, 1 ,and 3 , with of course x ? 1
I then pick an arbitrary value in each of the regions and determine the sign of the answer, the actual answer does not matter.
e.g let x = -5
(-)(-)/(-1) < 0 , so x <-2 works
let x = 0 , a value between -2 and 1
(-3)(+).(-) > 0 , no good
let x = 2 , a value between 1 and 3
(-)(+)/(+) < 0 , good
let x = 5 , a value > 3
(+)(+)/(+) > 0 , no good
so
x < -2 OR 1 < x < 3
so for this: (-)(-)/(-1) < 0 , so x <-2 works
you replace the 0 with -2 ? .
Also for this part let x = 5 , a value > 3
(+)(+)/(+) > 0 , no good
you had to change the sign because it was all positive?
To solve the inequality ((x-3)(x+2))/(x-1) < 0, you can follow these steps:
Step 1: Find the critical points
To determine where the expression is equal to zero or undefined, set both the numerator and the denominator equal to zero.
Setting the numerator equal to zero:
(x - 3)(x + 2) = 0
Setting the denominator equal to zero:
x - 1 = 0
Solving these equations, we find the critical points:
For the numerator: x = 3, x = -2
For the denominator: x = 1
So, the critical points are x = 3, x = -2, and x = 1.
Step 2: Create a sign chart
A sign chart helps us determine the signs of the expression ((x-3)(x+2))/(x-1) in different intervals.
Divide the number line into four intervals using the critical points:
Interval 1: (-∞, -2)
Interval 2: (-2, 1)
Interval 3: (1, 3)
Interval 4: (3, +∞)
Step 3: Determine the signs
Choose a test point from each interval (other than the critical points) and plug it into the expression ((x-3)(x+2))/(x-1). This will indicate the sign of the expression in that interval.
Let's choose the numbers -3, 0, 2 as test points.
Interval 1: (-∞, -2)
Plugging in -3: ((-3-3)(-3+2))/(-3-1) = (-6)(-1)/(-4) = 6/4 = 3/2 > 0 (positive)
Interval 2: (-2, 1)
Plugging in 0: ((0-3)(0+2))/(0-1) = (-3)(2)/(-1) = -6/-1 = 6 > 0 (positive)
Interval 3: (1, 3)
Plugging in 2: ((2-3)(2+2))/(2-1) = (-1)(4)/(1) = -4 > 0 (negative)
Interval 4: (3, +∞)
Plugging in 2: ((3-3)(3+2))/(3-1) = (0)(5)/(2) = 0 > 0 (positive)
Step 4: Interpret the sign chart
Now, we can analyze the sign chart to determine the values of x that satisfy the inequality ((x-3)(x+2))/(x-1) < 0.
From the sign chart, we can see that the expression is negative (less than zero) in the interval (-2, 1).
So, the solution to the inequality is: -2 < x < 1.