Find the range of real values of x for which (x-1)/(4x+5)<(x-3)/(4x-3).

Question

Find the range of real values of x for which \(\frac{x-1}{4x+5}<\frac{x-3}{4x-3}.\)

(a) \(\big(-∞,\frac{-5}{4}\big)∪\big(\frac34,∞\big)\)

(b) \(\big(\frac{-3}{4},\frac54\big)\)

(c) \(\big(-∞,\frac{-5}{4}\big]∪\big[\frac34,∞\big)\)

(d) \(\big(\frac{-5}{4},\frac34\big)\)

Answer 1

(d) \(\big(-\frac54,\frac34\big)\)

Putting each factor equal to zero, the critical points are \(-\frac54,\frac34.\)Plotting on the real number line, we have the following figure:

Thus the expression 16\(\big(x+\frac54\big)\big(x-\frac34\big)\)is positive when x < \(-\frac54,\,or\,\,x\,>\frac34.\)

∴  The required range = \(\big(-\frac54,\frac34\big)\).