Inequalities with Parameters
Inequalities with Parameters
Inequalities of the form ax > b, ax < b, where a and b are parameters, are called linear inequalities with a parameter.
Solving Linear Inequalities with a Parameter
Linear inequalities with a parameter have the form ax + b > 0. To solve them:
- Determine the values for which the parameter a equals zero.
- Consider separately the cases where a > 0 and a < 0.
Example: Solve (k − 3)x + 4 > 0.
- k − 3 = 0 → k = 3
- k − 3 > 0 → k > 3
- k − 3 < 0 → k < 3
Solving Quadratic Inequalities with a Parameter
Quadratic inequalities have the form ax² + bx + c > 0. Calculate the discriminant D = b² − 4ac and analyze the signs of the coefficients and discriminant.
Graphical Method
A graphical solution of an inequality f(x) ≥ g(x) means finding the values of x for which the graph of f(x) lies above the graph of g(x).
Example: Solve |x − 4| < cx.
Construct the graphs of y = |x − 4| and y = cx. The solution is the values of x where the V-shape lies below the line.
Answer: for c ≤ −1: x ∈ (−∞; 4/(1−c)); for −1 < c < 0: x ∈ (−4/(c+1); 4/(1−c)); for 0 ≤ c ≤ 1: no solutions; for c > 1: x ∈ (4/(1−c); +∞).