Inequalities of the form g(x) >= f(x) (HL)

Q: What is the general process for solving \(g(x) \geq f(x)\) using a Graphic Display Calculator (GDC)?

Graph both functions, find their \(x\)-coordinates of intersection, identify where the graph of \(g(x)\) is above \(f(x)\), and write the solution as an interval of \(x\)-values.

Q: What is a common mistake to avoid when stating the final solution to an inequality?

A common mistake is providing the \(y\)-coordinates of the intersection points; the solution must always be a set of \(x\)-values.

Q: What is the 'Single Function Method' for solving inequalities analytically?

Rearrange the inequality to \(g(x) - f(x) \geq 0\), effectively comparing a single new function \(h(x)\) against the \(x\)-axis.

Q: In an analytical approach, where do the 'Critical Values' (boundaries) of an inequality occur?

Critical values occur at the zeros of the function (where \(h(x) = 0\)) and at points where the function is undefined (vertical asymptotes).

Q: How is a sign diagram used to solve an inequality?

Place critical values on a number line, then test a value from each resulting interval in the function to see if the output is positive or negative.

Q: Why is it forbidden to cross-multiply by a variable expression in an inequality?

Because the sign of the variable expression is unknown; if it is negative, the inequality sign must flip, but if it is positive, it stays the same.

Q: What is the 'Square the Denominator' trick for rational inequalities?

Multiply both sides by the square of the denominator, \([f(x)]^2\), which is guaranteed to be positive and thus preserves the inequality sign.

Q: How do you analytically solve a modulus inequality of the form \(|g(x)| \geq f(x)\)?

Split the inequality into two separate cases: \(g(x) \geq f(x)\) and \(-g(x) \geq f(x)\) (which is equivalent to \(g(x) \leq -f(x)\)).

Mathematics - Analysis and Approaches · IB Diploma Programme (DP) - SL & HL · 5 min read · Updated: 7 Sept 2026

Introduction to Solving HL Inequalities

Welcome to one of the most practical chapters in the Functions topic! In earlier math, you likely solved simple inequalities like \(2x + 3 > 7\). At the Higher Level (HL), we take this a step further by comparing two different functions: \(g(x) \geq f(x)\). Essentially, we are trying to find the values of \(x\) where the graph of one function sits on top of or intersects the graph of another. Whether you are using your Graphic Display Calculator (GDC) or solving it by hand with algebra, the goal remains the same: finding the "winning" intervals for \(g(x)\).

Understanding the Visual Concept

Before diving into the algebra, let’s think about this graphically.

When we see the statement \(g(x) \geq f(x)\), we are looking for the domain (the \(x\)-values) where the curve of \(g(x)\) is higher than the curve of \(f(x)\) on a coordinate plane.

Analogy: The Mountain Climbers Imagine two people, Greg (\(g\)) and Frank (\(f\)), climbing different paths on a mountain. The inequality \(g(x) \geq f(x)\) is asking: "At which horizontal positions is Greg higher up than Frank, or standing at the exact same height?"

Quick Review: The Inequality Signs

\(g(x) > f(x)\): \(g\) is strictly above \(f\).
\(g(x) < f(x)\): \(g\) is strictly below \(f\).
\(g(x) \geq f(x)\): \(g\) is above or touching \(f\).
\(g(x) \leq f(x)\): \(g\) is below or touching \(f\).

Method 1: The Graphical Approach (Paper 2 & 3)

Since technology is required for Paper 2 and Paper 3, the GDC is your best friend for these questions.

Step-by-Step GDC Process:

Graph both functions: Input \(y_1 = g(x)\) and \(y_2 = f(x)\) into your calculator.
Find the points of intersection: Use the "Intersect" tool (usually under the CALC or G-Solve menu) to find the \(x\)-coordinates where \(g(x) = f(x)\).
Identify the intervals: Look at the screen. In which sections is the line/curve for \(g(x)\) physically above \(f(x)\)?
Write the solution: Express these sections using interval notation (e.g., \([a, b]\)) or inequality notation (e.g., \(a \leq x \leq b\)).

Common Mistake: Students often give the \(y\)-coordinate in their final answer. Remember, the solution to an inequality is always a set of \(x\)-values!

Key Takeaway: On Paper 2, always sketch the graphs you see on your screen onto your exam paper to show your method. Label the intersection points clearly!

Method 2: The Analytical Approach (Paper 1)

On Paper 1, you won't have a calculator. You need a reliable algebraic strategy. The most robust method is the Single Function Method.

1. Move everything to one side

Rewrite the inequality so that one side is zero:

\(g(x) - f(x) \geq 0\)

Let's call this new combined function \(h(x)\). Now you are just looking for where \(h(x)\) is positive or zero.

2. Find the Critical Values

Critical values are the "boundaries" where the inequality might change from true to false. These occur at:

Zeros: Where \(h(x) = 0\) (The \(x\)-intercepts).
Undefined points: Where \(h(x)\) has vertical asymptotes (usually where the denominator is zero).

3. Create a Sign Diagram

Draw a number line and mark all your critical values. This divides the number line into test intervals. Pick a number from each interval and plug it into \(h(x)\) to see if the result is positive (+) or negative (-).

4. State the Final Interval

Since we want \(h(x) \geq 0\), choose the intervals marked with a (+).
Example: If you are solving \(\frac{1}{x} \geq x\), you would rearrange to \(\frac{1}{x} - x \geq 0\). The critical values are \(x = 0\) (undefined) and \(x = 1, x = -1\) (zeros). Testing the zones would show the inequality holds for \(x \in (-\infty, -1] \cup (0, 1]\).

Special Case: Rational Inequalities

Warning! Never cross-multiply by a variable expression in an inequality. For example, in \(\frac{g(x)}{f(x)} \geq 1\), you cannot simply multiply by \(f(x)\) because you don't know if \(f(x)\) is positive or negative. If it's negative, the inequality sign must flip!

The "Square the Denominator" Trick

If you absolutely must multiply to clear a fraction, multiply both sides by the square of the denominator: \([f(x)]^2\). Since a square is always positive, the inequality sign is guaranteed to stay the same.

\(\frac{g(x)}{f(x)} \geq k \implies g(x) \cdot f(x) \geq k \cdot [f(x)]^2\)

Solving with Modulus Functions

Sometimes you will face inequalities like \(|g(x)| \geq f(x)\).

Graphically: Sketch the modulus graph (where any part below the \(x\)-axis is reflected upward) and see where it is above \(f(x)\).
Analytically: This usually requires solving two cases: \(g(x) \geq f(x)\) and \(-g(x) \geq f(x)\).

Cross-reference: For more on sketching these, see the chapter on "Modulus, reciprocal and composite graphs".

Quick Summary Checklist

1. Boundaries Matter: Did you include the endpoints? Use \([\text{ or } ]\) for \(\geq\) and \((\text{ or } )\) for \(>\).
2. Asymptotes are Exclusive: Even in a \(\geq\) inequality, the value where a function is undefined must always have a curved bracket \(()\) because the function can never actually reach that value.
3. Sign Flips: Did you multiply or divide by a negative number? If so, flip that sign!
4. Test your points: Always double-check by picking a simple number like \(x=0\) or \(x=1\) to see if your final interval makes sense.

Key Takeaway Summary

Solving \(g(x) \geq f(x)\) is about finding the shared horizontal space where the height of \(g\) is greater than or equal to the height of \(f\). Use your GDC to find intersections on Paper 2; use a sign diagram and critical values on Paper 1. Never cross-multiply by variables!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the visual interpretation of the inequality \(g(x) \geq f(x)\) on a coordinate plane?

It represents the set of \(x\)-values (domain) where the graph of \(g(x)\) is either above or intersecting the graph of \(f(x)\).

What are the four main inequality signs and their meanings in terms of graph positions?

\(>\) (strictly above), \(<\) (strictly below), \(\geq\) (above or touching), and \(\leq\) (below or touching).

What is the general process for solving \(g(x) \geq f(x)\) using a Graphic Display Calculator (GDC)?

Graph both functions, find their \(x\)-coordinates of intersection, identify where the graph of \(g(x)\) is above \(f(x)\), and write the solution as an interval of \(x\)-values.

What is a common mistake to avoid when stating the final solution to an inequality?

A common mistake is providing the \(y\)-coordinates of the intersection points; the solution must always be a set of \(x\)-values.

What is the 'Single Function Method' for solving inequalities analytically?

Rearrange the inequality to \(g(x) - f(x) \geq 0\), effectively comparing a single new function \(h(x)\) against the \(x\)-axis.

In an analytical approach, where do the 'Critical Values' (boundaries) of an inequality occur?

Critical values occur at the zeros of the function (where \(h(x) = 0\)) and at points where the function is undefined (vertical asymptotes).

How is a sign diagram used to solve an inequality?

Place critical values on a number line, then test a value from each resulting interval in the function to see if the output is positive or negative.

Why is it forbidden to cross-multiply by a variable expression in an inequality?

Because the sign of the variable expression is unknown; if it is negative, the inequality sign must flip, but if it is positive, it stays the same.

What is the 'Square the Denominator' trick for rational inequalities?

Multiply both sides by the square of the denominator, \([f(x)]^2\), which is guaranteed to be positive and thus preserves the inequality sign.

How do you analytically solve a modulus inequality of the form \(|g(x)| \geq f(x)\)?

Split the inequality into two separate cases: \(g(x) \geq f(x)\) and \(-g(x) \geq f(x)\) (which is equivalent to \(g(x) \leq -f(x)\)).

Why must vertical asymptotes always use curved brackets (exclusive notation) in an inequality solution?

Because a function is undefined at an asymptote, it can never actually reach or 'equal' that value, regardless of whether the inequality is \(\geq\) or \(>\).

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