can somebody tell me how to solve this inequality equation?
(a-2)(a-3)<0
I know the opposite was 2 and 3..but how do i determine the greater than, less than or equal to? Explain please.
Eq1. Given: (x - 2) (x - 3) < 0.
Eq2. Graph (x - 2) (x - 3) = 0,
x - 2 = 0,
x = 2 = x-Int.
x - 3 = 0,
x = 3 = x-Int.
Find the coordinates of the vertex:
h = Ave of the x-Ints,
h = Xv = (2 + 3) / 2 = 2.5.
Substitute 2.5 for x in Eq2 to get
y coordinate of vertex:
k = Yv = (2.5 - 2) (2.5 - 3) = -0.25.
V(h , k) = V(2.5 , -0.25).
Use the coordinates of the vertex and x-ints to sketch the graph:
(2 , 0) , V(2.5 , -0.25) , (3 , 0).
Your inequality uses the < sign.
So all points on the graph between the
X-Ints are negative(less than 0) and
satisfy the inequality.
In Eq form: 2 < X < 3.
The Eq states that all values of x >
2 but < 3 satisfy the inequality.
You can use the letter a as your variable if you desire. I didn't mean to use x.
To solve this inequality equation, (a-2)(a-3)<0, you need to find the values of a that make the inequality true.
First, let's understand the concept of factors and signs. In this equation, (a-2) and (a-3) are the factors. The factor (a-2) represents the difference between "a" and 2, and the factor (a-3) represents the difference between "a" and 3.
To determine the sign of the product (a-2)(a-3), you need to consider the signs of each factor.
Now, looking at the first factor, (a-2), we know that when "a" is greater than 2, the factor (a-2) is positive. Conversely, when "a" is less than 2, the factor (a-2) is negative.
Next, considering the second factor, (a-3), we can apply the same logic. When "a" is greater than 3, the factor (a-3) becomes positive. When "a" is less than 3, the factor (a-3) is negative.
To find when the product (a-2)(a-3) is less than zero (negative), we need to analyze the signs of the factors.
1. When both factors are positive, the product is positive.
2. When both factors are negative, the product is positive.
3. When one factor is positive and the other is negative, the product is negative.
Now, to determine when the product (a-2)(a-3) is negative, we need to find the values of "a" that make the inequality true.
Case 1: Both factors are positive:
In this case, (a-2)>0 and (a-3)>0.
Solving these inequalities, we get a>2 and a>3. However, since "a" cannot be greater than both 2 and 3 at the same time, this case is not possible.
Case 2: Both factors are negative:
In this case, (a-2)<0 and (a-3)<0.
Solving these inequalities, we get a<2 and a<3. Again, we cannot find values of "a" that satisfy both conditions simultaneously, so this case is also not possible.
Case 3: One factor is positive, and the other factor is negative:
In this case, one factor must be greater than zero, and the other factor must be less than zero.
Consider (a-2)>0 (positive) and (a-3)<0 (negative).
This implies a>2 and a<3.
To summarize, the solution to the inequality equation (a-2)(a-3)<0 is a>2 and a<3. In other words, a must be greater than 2 and less than 3 to satisfy the inequality.