First, factor the numerator and denominator of the expression as:
((x -3)(x + 4))/((2x + 3 )(x - 3))
Next, cancel the common terms from the numerator and denominator:
(color(red)(cancel(color(black)((x -3))))(x + 4))/((2x + 3 )color(red)(cancel(color(black)((x -3))))) =>
(x + 4)/(2x + 3)
However, because we cannot divide by 0 we must ensure:
2x + 3 != 0 and x - 3 != 0
Or
Condition 1:
2x + 3 != 0
2x + 3 - color(red)(3) != 0 - color(red)(3)
2x + 0 != -3
2x != -3
(2x)/color(red)(2) != -3/color(red)(2)
(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) != -3/2
x != -3/2
Condition 2:
x - 3 != 0
x - 3 + color(red)(3) != 0 + color(red)(3)
x - 0 != 3
x != 3
Therefore, the simplified expression is:
(x + 4)/(2x + 3) Where x != -3/2 and x != 3
