Answer:

Solve Rational Inequalities Examples With Solutions. Rational Inequalities are solved in the examples below. Knowing that the sign of an algebraic expression changes at its zeros of odd multiplicity, solving an inequality may be reduced to finding the sign of an algebraic expression within intervals defined by the zeros of the expression in question.

Step-by-step explanation:

Questions a)

solve inequality question

x-2/x+1 > 0

Solution

We first arrange the zeros of the numerator and the denominator, of the rational expression on the left of the inequality symbol, on the number line, from smallest to the largest as follows.

-∞ - 1 2 +∞

Select a value of x in any of the intervals and use it to find the sign of the rational expression.

Example for x = -3 in the interval (-∞ , -1), the rational expression (x - 2)/(x + 1) = (- 3 - 2)/(- 3 + 1) = 5 / 2. Hence the rational expression (x - 2)/(x + 1) is positive on the interval (-∞ , -1) .  

-∞ + - 1               2 +∞

The zeros -1 and 2 are of odd multiplicity and therefore the sign of the expression (x - 2)/(x + 1) will change at both zeros as we go from on interval to another. Hence the signs of the expression (x - 2)/(x + 1) as we go from left to right are

-∞ + - 1 - 2 + +∞

The solution set of the inequality is given by the union of all intervals where (x - 2)/(x + 1) is positive or equal to 0. Hence the solution set for the above inequality, in interval notation, is given by:

(-∞ , -1) ∪ [ 2 , +∞)