Introduction to Inequalities of the Form g(x) ≥ f(x)

In AHL 2.15, you are expected to solve inequalities that compare two functions , finding the values of $x$ where one function is greater than, less than, or equal to the other.

Inequality of the Form g(x) ≥ f(x): An inequality that asks: for which values of $x$ is $g (x)$ greater than or equal to $f (x)$ ? The solution is a set of $x$ -values (often an interval or union of intervals) satisfying the condition.

There are two main strategies for solving these inequalities:

Graphical approach , sketch both functions and identify where one lies above the other
Analytical approach , rearrange algebraically to isolate the variable

Both methods have their place. The graphical approach builds intuition and is powerful for complex functions; the analytical approach produces exact solutions and is essential in non-calculator settings.

Note

IB exams may require either method depending on the question. If a GDC is permitted, graphical methods are often faster. If not, algebraic techniques are essential.

Graphical Approach

The graphical method involves plotting both $y = g (x)$ and $y = f (x)$ on the same set of axes, then identifying the $x$ -values where the graph of $g (x)$ lies above or on the graph of $f (x)$ .

Steps:

Sketch (or use a GDC to plot) both functions on the same axes.
Find the intersection points , these are the boundaries of your solution.
Identify the regions where $g (x) \geq f (x)$ , i.e., where the graph of $g$ is on top of or touching $f$ .

Example

Solve $x^{2} + 1 \geq 2 x$ graphically.

Step 1: Plot $y = x^{2} + 1$ (an upward-opening parabola with vertex at $(0, 1)$ ) and $y = 2 x$ (a straight line through the origin with slope 2).

Step 2: Find where they intersect by setting $x^{2} + 1 = 2 x$ , giving $x^{2} - 2 x + 1 = 0$ , so $(x - 1)^{2} = 0$ , meaning $x = 1$ is a repeated root , the line is tangent to the parabola at $x = 1$ .

Step 3: Since the parabola opens upward and touches the line only at $x = 1$ , the parabola lies on or above the line for all $x$ .

$Solution: x \in R (all real numbers)$

Exam Tip

The intersection points of $g (x)$ and $f (x)$ are always the boundary values of your solution set. Always check whether the boundary is included ( $\geq$ or $\leq$ ) or excluded ( $>$ or $<$ ).

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Introduction to Inequalities of the Form g(x) ≥ f(x)

In AHL 2.15, you are expected to solve inequalities that compare two functions , finding the values of $x$ where one function is greater than, less than, or equal to the other.

There are two main strategies for solving these inequalities:

Graphical approach , sketch both functions and identify where one lies above the other
Analytical approach , rearrange algebraically to isolate the variable

Note

IB exams may require either method depending on the question. If a GDC is permitted, graphical methods are often faster. If not, algebraic techniques are essential.

Graphical Approach

Steps:

Sketch (or use a GDC to plot) both functions on the same axes.
Find the intersection points , these are the boundaries of your solution.
Identify the regions where $g (x) \geq f (x)$ , i.e., where the graph of $g$ is on top of or touching $f$ .

Example

Solve $x^{2} + 1 \geq 2 x$ graphically.

Step 1: Plot $y = x^{2} + 1$ (an upward-opening parabola with vertex at $(0, 1)$ ) and $y = 2 x$ (a straight line through the origin with slope 2).

Step 3: Since the parabola opens upward and touches the line only at $x = 1$ , the parabola lies on or above the line for all $x$ .

$Solution: x \in R (all real numbers)$

Exam Tip

The intersection points of $g (x)$ and $f (x)$ are always the boundary values of your solution set. Always check whether the boundary is included ( $\geq$ or $\leq$ ) or excluded ( $>$ or $<$ ).

Free preview

AHL 2.15—Solutions of inequalities

Introduction to Inequalities of the Form g(x) ≥ f(x)

Graphical Approach

9 more sections in this topic

AHL 2.15—Solutions of inequalities

Introduction to Inequalities of the Form g(x) ≥ f(x)

Graphical Approach

9 more sections in this topic