ALGEBRA - solved problems
Rational Inequalities
 94
 Solve the rational inequality and draw the graph of the rational function.
Solution:   The solution set of the given rational inequality includes all numbers x which make the inequality greater then or equal to 0, or which make the sign of the rational expression to be positive or 0.
A rational expression is positive if both the numerator and the denominator are positive or if both are negative, and the rational expression equals 0 when its numerator is equal to 0 that is therefore, we have to solve two simultaneous inequalities: The solutions represented on the number line are shown below. Thus, the solution set of the given inequality written in the interval notation is (- oo, -1) U [2, oo ).
The graph of the given rational function is translated equilateral (or rectangular) hyperbola.
 A rational function of the form can be rewritten into where the vertical asymptote, the horizontal asymptote
 and the parameter Therefore, values of the vertical and the horizontal asymptote correspond to the coordinates of the horizontal and the vertical translation of the source equilateral hyperbola  y = k/x, respectively.
 Thus, given rational function where, a = 1, b = -2  and  c = 1, d = 1
 has the vertical asymptote the horizontal asymptote and the parameter Therefore, its source function is the equilateral or rectangular hyperbola The graph of given rational function is shown below, (see the above example of the rational inequality). 95
Solve the rational inequality Solution:   The solution set of the given rational inequality includes all numbers x which make the inequality less then or equal to 0, or which make the sign of the rational expression to be negative or 0.
A rational expression is negative if the numerator and the denominator have different signs, and the rational expression equals 0 when its numerator is equal to 0 that is, therefore, we have to solve two simultaneous inequalities.
We graph the numerator and the denominator in the same coordinate system to find all points of the x-axis that satisfy given inequality.
The zero points of the numerator and the denominator divide the x-axis into four intervals at which given rational expression changes sign.
 x2 + 2x - 3 = 0,  a = 1, b = 2 and c = -3  The solutions of the two pairs of the simultaneous inequalities are intersections of sets of their partial solutions, as is shown below. Therefore, the solution set is ( - oo, -3] U ( -2, 1].
Absolute value inequalities
Solving linear inequalities with absolute value
Graphical interpretation of the definition of the absolute value of a function = f(x) will help us solve linear inequality with absolute value.
 96 Solve the absolute value inequality  | x - 2 | < 3.
Solution:   Graphical interpretation of the given inequality will help us find values of x for which left side of the inequality is less then or equal to 3.
For values of x for which y is nonnegative, the graph of | y | is the same as that of  = x - 2.
For values of x for which y is negative, the graph of | y | is a reflection of the graph of y across the x axis.
Since the graph of  = x - 2 has y negative on the interval (- oo, 2) it is this part of the graph that has to be reflected on the x axis. The graph shows that values of x from the closed interval [-1, 5] satisfy the given inequality.
The same result can be obtained algebraically by solving the compound inequality.  x Î [-1, 5]
 97 Solve the absolute value inequality  | x - 2 | > 3.
Solution:   Again, graphical interpretation of the given inequality will help us find values of x for which left side of the inequality is greater than 3.
The graph shows that values of x from the open intervals, (- oo, -1) or (5, oo) satisfy the given inequality. The same result can be obtained algebraically by solving the two inequalities    Solved problems contents