Solve the inequality.
[( 2x^2+5x-3 ) / (x^2 -8x+ 16) ] >= 0
I factored out the bottom to (X-4)(X-4). The top should be 2(x+ ? )(x- ?). What would I do next to finish the problem?
the top factored is (2x-1)(x+3) using the grouping method.
With the top and bottom factored out, there is nothing to cancel out. So what do i do from here?
To solve the inequality, you first correctly factored the denominator to (x-4)(x-4). Now, let's factor the numerator further:
2x^2 + 5x - 3 = 2(x+ ? )(x- ? )
To find the missing terms, we can use the fact that the coefficient of the middle term, 5x, is the sum of the products of the two numbers we are looking for. In this case, we are looking for two numbers that multiply to give -3 and add up to 5.
To factorize -3, we can have (-1)(3) or (1)(-3). By trial and error, we find that (-1)(3) will give us the sum of 5.
So, the factored form of the numerator becomes:
2(x - 1)(x + 3)
Now, we have the inequality:
[2(x - 1)(x + 3)] / [(x - 4)(x - 4)] ≥ 0
To solve this inequality, we need to consider the sign of each factor separately.
1. (x - 1)
2. (x + 3)
3. (x - 4)
To do this, we can create a sign table where we consider each factor separately and determine when it changes sign:
(x - 1) | (x + 3) | (x - 4)
----------------|----------------|--------------
x < 1 | x > -3 | x < 4
----------------|----------------|--------------
Now, we need to determine when the entire expression is positive or zero, which includes cases where the expression equals zero:
1. When all three factors are positive:
x > 4
2. When one factor is positive and the other two are zero:
1 < x < 4
Thus, the inequality solution is the combination of these two intervals:
x > 4 or 1 < x < 4