Basics of Graphical Inequalities

A graphical inequality represents a region of a graph, rather than a specific line or point.
Inequalities are mathematical expressions involving less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥) signs.
These inequalities allow for a range of solutions, rather than one specific answer in a given problem.

Core Concepts

Graphs can be used to visualise and solve inequality problems.
For linear inequalities, the inequality line is usually dotted for < and > (since the points on the line are not included in the solution), and solid for ≤ and ≥ (as points on the line are included).
To represent a ‘greater’ inequality on a graph, you shade above the line. For a ‘less’ inequality, you shade below the line.
An x > a inequality creates a vertical line that shades to the right. Similarly, x < a forms a vertical line but shades to the left.
An y > a inequality results in a horizontal line that shades upwards, and y < a forms a horizontal line but shades downwards.

Often in GCSE problems, you will be asked to find the region that satisfies a set of inequalities. This involves shading the regions for each inequality and finding where they intersect.
The intersection of regions, usually highlighted by a different colour, represents the combined solution to the inequalities.

Graphing inequalities becomes more complex with quadratic inequalities, where the graph may be a curve rather than a straight line.
When solving quadratic inequalities, think of the shaded region as being inside or outside the curve, rather than above or below a line.

Always start with graphing the equal form of the equation first, then apply the inequality shading rules.
While solving graphical inequalities, keep in mind that there can be multiple or no solutions based on the given conditions.
Check your solutions by substituting them into the original inequality to ensure they satisfy the inequality conditions.

With consistent practice of graphical inequalities, you can improve your problem solving abilities and build robust mathematical reasoning skills.