Solving inequalities and shading their solution regions
A linear inequality compares two expressions using <, >, ≤ or ≥, such as 3x − 5 < 7. It is solved like an equation, with one important difference: when both sides are multiplied or divided by a negative number, the direction of the inequality sign must be reversed. So −2x > 6 gives x < −3, not x > −3. The solution is usually a whole range of numbers, which can be written in interval notation, such as (−∞, 4), or shown on a number line with an open circle for a strict sign and a filled circle for ≤ or ≥.
An inequality in two variables, such as 2x + 3y ≤ 12, is shown on a graph. First draw the boundary line 2x + 3y = 12, using a solid line when the sign includes equality and a dashed line when it does not. Then test a point that is not on the line, usually the origin: since 0 ≤ 12 is true, the half-plane containing the origin is shaded. For a system of inequalities, the solution is the region where all the shaded half-planes overlap.
Why it matters
Inequalities describe limits and constraints, such as a budget that cannot be exceeded, a minimum mark needed to pass, or the safe load on a bridge. Systems of inequalities are the basis of linear programming in Class 12, which businesses use to decide how much of each product to make for the greatest profit. They also appear when finding the domain of functions in calculus.
Key terms
- Inequality
- A statement that one expression is greater than, less than, or not equal in a stated way to another.
- Interval notation
- A way of writing a range of numbers using brackets, such as [2, 5) or (−∞, 4).
- Boundary line
- The line obtained by replacing the inequality sign with an equals sign, which separates the plane into two halves.
- Half-plane
- Either of the two regions into which a straight line divides the coordinate plane.