3.6PureCore

Inequalities on Graphs

Inequalities can be read straight off a graph. Where one curve sits above another, the corresponding inequality holds. This turns a tricky algebra problem into reading intervals from a sketch.

25 min Video by Zeeshan Zamurred Equations and Inequalities

Edexcel AS level Maths: 3.6 Inequalities on GraphsWatch the full walkthrough before the notes below.

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What you'll be able to do

Interpret f(x) > g(x) as “where one graph is above the other”
Use intersection points to define the intervals
Solve inequalities graphically
Connect the graph regions to algebraic solutions

Above and below

The statement $f (x) > g (x)$ is true for the $x$ -values where the graph of $f$ lies $above$ the graph of $g$ . Likewise $f (x) < g (x)$ is where $f$ is below $g$ .

Intersections set the boundaries

The points where the two graphs cross are the boundaries of the solution intervals. Solve $f (x) = g (x)$ to find them, then decide which intervals satisfy the inequality.

1Find intersections:

x^{2} = x + 2 \Rightarrow x^{2} - x - 2 = 0 \Rightarrow (x - 2) (x + 1) = 0

2Boundaries at

x = - 1

and

x = 2

3The parabola is below the line between these points.

Answer

- 1 < x < 2

Tip — Find the crossing points first — they always divide the x-axis into the candidate intervals.

Reading the right region

Once you have the boundaries, a quick sketch (or testing a value in each interval) tells you which side the inequality holds.

Formula recap

f (x) > g (x) \Leftrightarrow f above g

Graphical meaning.

f (x) = g (x) \Rightarrow boundary x -values

Intersections set the limits.

Common mistakes to avoid

Choosing the wrong region after finding the boundaries.

Test a value from each interval, or sketch, to confirm which side holds.

Ignoring that both graphs matter, not just one.

Compare the two graphs — it is about which is above the other.

Key takeaways

f(x) > g(x) means f is above g on the graph.
Intersection points are the interval boundaries.
Sketch or test points to pick the correct region.

Test yourself

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