I am working on solving this inequality and could really use some help.

Question

(x-3)/(x+4)>2
Thanks :)

Answer 1

The rational inequality is (x - 3)/(x + 4) > 2.

write the inequality in general form with the rational expression on the left and zero on the right.

[(x - 3)/(x + 4)] - 2 > 0

[(x - 3) - 2(x + 4)]/(x + 4) > 0

(- x - 11)/(x + 4) > 0

(x + 11)/(x + 4) < 0

The zeros (the x-values for which its numerator is zero) and its undefined values (the x-values for which its denominator is zero) of the rational expression are called key numbers.

Numerator is zero, x + 11 = 0 ⇒ x = - 11.

Denominator is zero, x + 4 = 0 ⇒ x = - 4.

The key numbers are x = - 11 and x = - 4. So, the polynomial’s test intervals are (-∞, -11), (- 11, - 4) and (- 4, ∞).

In each test interval, choose a representative x-value and evaluate the polynomial.

Test Interval   x-value   Polynomial Value [(x + 11)/(x + 4) < 0]     Conclusion

(-∞, -11)            x = -12    [(-12) + 11]/[(-12) + 4] = (-1)/(-8) = 1/8 > 0        Positive

(-11, -4)             x = -5      [(-5) + 11]/[(-5) + 4] = 6/(-1) = -6 < 0                 Negative

(-4, ∞)               x = 0       [(0) + 11]/[(0) + 4] = 11/4 = > 0                          Positive

From this we can conclude that the inequality is satisfied on the open intervals (-11, -4). So, the solution set is (- 11, - 4). Note that the original inequality contains a “less than” symbol. This means that the solution set does not contain the endpoints of the test interval (- 11, - 4).