Question 1120102
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I will assume that both the (x-2) and (x-4) factors are in the denominator: {{{((2x-3))/((x-2)(x-4))}}}<br>
The critical points are where the numerator or denominator is zero.<br>
The function value is zero when the numerator is zero:
2x-3 = 0 -->  x = 1.5<br>
The function value is undefined when the denominator is zero:
x-2 = 0 -->  x = 2;  and x-4 = 0 -->  x = 4<br>
The critical points divide the domain into intervals; you can use test points and do a sign analysis to determine whether the function value is positive or negative on each interval.  An even number of negative factors means the function value is positive; an odd number means the function value is negative:<br>
x < 1.5: 3 negative factors; function value negative
x = 1.5: function value 0
1.5 < x < 2: 2 negative factors; function value positive
x = 2: function value undefined
2 < x < 4: 1 negative factor; function value negative
x = 4: function value undefined
x > 4: 0 negative factors; function value positive<br>
Note you can speed up the process without using test points in each interval.  Simply start at one end of the number line and "walk" towards the other end, noting that the sign of the function changes each time you pass a critical point:<br>
function value negative for large negative values of x (3 negative factors)
function value changes to positive when we pass x = 1.5
function value changes to negative when we pass x = 2
function value changes to positive when we pass x = 4<br>
Answer: The function value is negative for x < 1.5 and for 2 < x < 4<br>
{{{graph(400,400,-2,6,-10,10,(2x-3)/((x-2)(x-4)))}}}