Linear and quadratic inequalities (Higher Tier)

Understanding Linear and Quadratic Inequalities

Similar to equations, inequalities express a relationship between values, but rather than expressing an exact equivalence, inequalities express ‘greater than’, ‘less than’, ‘greater than or equal to’, or ‘less than or equal to’ relationships.
Linear inequalities can be represented algebraically, such as x < 2, x > 7, or 4x ≤ 8. They relate two expressions as being unequal, with a linear relationship.
Quadratic inequalities, on the other hand, relate a quadratic function and a linear function or another quadratic function. They take the form of ax^2 + bx + c < 0, ax^2 + bx + c ≤ 0, ax^2 + bx + c > 0, or ax^2 + bx + c ≥ 0.

Solving Linear and Quadratic Inequalities

Solve linear inequalities just as you would a linear equation, but remember that if you multiply or divide both sides by a negative number, the inequality sign needs to be reversed.
For quadratic inequalities, rewrite the inequality in standard form (i.e. set it to 0) and solve for x as if it was an equation. The solutions, or roots, will split the x-axis into intervals.
To find regions of x that satisfy the inequality, select a test point from each interval and substitute it back into the inequality. If the original inequality is satisfied, the entire interval is a valid solution.

Graphing Linear and Quadratic Inequalities

Linear inequalities are represented on a graph as a straight line with a shaded region that represents the solution set.
When the inequality sign involves ‘equal to’, the line is solid because it includes the points on the line. When the sign does not, the line is dashed to indicate that points on the line are not part of the solution.
Quadratic inequalities are generally represented by a parabola. The shaded region that satisfies the inequality can either be above or below the curve, or between parts of the curve based on whether the inequality is ‘less than’, ‘greater than’, ‘less than or equal to’, or ‘greater than or equal to’.

Understanding Solution Sets of Linear and Quadratic Inequalities

For a linear inequality, the solution set can be an interval, either bounded (e.g. 1 < x < 5) or unbounded (e.g. x > 5).
For a quadratic inequality, the solution set can either be a single interval (like linear inequalities) or two separate intervals. For example, if a quadratic equation results in x^2 - 5x + 6 < 0, the roots are x = 2 and x = 3. The solution set for the inequality could involve two separate intervals, like x < 2 or x > 3, or a single interval like 2 < x < 3. This depends on the original inequality sign.