Solving Inequalities of the Form g(x) ≥ f(x)
In Mathematics AA HL, students are expected to solve inequalities of the form g(x) ≥ f(x), where g(x) and f(x) are functions. This type of inequality compares two functions and asks for the values of x where one function is greater than or equal to the other.
Graphical Approach
One method to solve g(x) ≥ f(x) is by graphing both functions and identifying the regions where g(x) is above or on f(x).
ExampleConsider the inequality $x^2 + 1 \geq 2x$. To solve this graphically:
- Graph $y = x^2 + 1$ (a parabola)
- Graph $y = 2x$ (a straight line)
- Identify where the parabola is above or touching the line
The solution is the x-values where the parabola is above or touching the line, which in this case is $x \leq -1$ or $x \geq 2$.
TipWhen graphing, pay attention to the points of intersection between g(x) and f(x). These points often represent the boundaries of the solution set.
Analytical Approach
The analytical method involves algebraically manipulating the inequality to solve for an exact value of $x$.
- Rearrange the inequality to have zero on one side: g(x) - f(x) ≥ 0
- Solve this new inequality using algebraic techniques, such as factoring
