Inequalities: Representing solution sets on number lines

Algebra • Solving equations and inequalities

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How to test sign intervals for a quadratic inequality?

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Find the roots, split the number line into intervals by roots, choose a test value from each interval, and evaluate sign.

Key concepts

What you'll likely be quizzed about

Inequality symbols and solution sets

An inequality uses <, >, ≤ or ≥ to compare expressions. The symbol < or > indicates strict comparison and excludes boundary values. The symbol ≤ or ≥ indicates inclusive comparison and includes boundary values. A solution set is the collection of all values that satisfy the inequality.

Solving linear inequalities in one variable

Linear inequalities follow the same algebraic steps as linear equations: simplify both sides, collect like terms and isolate the variable. Because multiplication or division by a negative number reverses the inequality sign, sign reversal occurs when multiplying or dividing both sides by a negative. When the expression simplifies to a true statement with no variable (for example 3 < 5), the solution set is all real numbers; when it simplifies to a false statement (for example 2 > 7), there is no solution.

Solving linear inequalities in two variables

A linear inequality in two variables produces a half-plane on the coordinate plane. Converting to the form y = mx + c shows the boundary line. A strict inequality (< or >) produces a dashed boundary line to show exclusion of the boundary. An inclusive inequality (≤ or ≥) produces a solid boundary line to show inclusion. A test point determines which side of the boundary contains the solutions; if the test point satisfies the inequality, the side containing that point is the solution region.

Solving quadratic inequalities in one variable

A quadratic inequality requires finding boundary points where the quadratic equals zero. Factorisation or the quadratic formula finds these roots. The number line splits into intervals determined by the roots. Testing one value from each interval shows whether the quadratic expression is positive or negative on that interval. The solution set consists of intervals where the inequality sign holds, including roots when the inequality is non-strict (≤ or ≥) and excluding roots when strict (< or >).

Number line, interval and set notation

A number line represents continuous solution sets visually using open or closed dots and arrows. An open dot indicates exclusion of a boundary; a closed dot indicates inclusion. Interval notation uses parentheses for exclusion and brackets for inclusion, for example (−∞, 3) or [2, 5). Set-builder notation states conditions explicitly, for example {x | x > 4}. Each form communicates the same solution set in a different, standard format.

Graphical representation and boundary conventions

Graphs for one-variable inequalities use number line conventions. Graphs for two-variable inequalities shade the solution region on the plane. A dashed boundary line indicates that points on the line do not satisfy the inequality. A solid boundary line indicates that points on the line satisfy the inequality. Shading direction follows from testing a point or by analysing the inequality algebraically.

Key notes

Important points to keep in mind

Multiplying or dividing by a negative number reverses the inequality sign.

Use an open dot for strict inequalities and a closed dot for inclusive inequalities on number lines.

Convert two-variable linear inequalities to y = mx + c to identify the boundary line and shading direction.

Find quadratic roots to split the number line into test intervals for sign analysis.

Use interval notation with ( or ) for exclusion and [ or ] for inclusion of endpoints.

Test a single point in a region to confirm whether the region satisfies the inequality.

Represent x ≠ a as the union of two intervals: (−∞, a) ∪ (a, ∞).

A true statement without variables indicates all real numbers; a false statement indicates no solution.

Shade the half-plane that makes the inequality true; use dashed boundary for < or > and solid for ≤ or ≥.

When solving chained inequalities, treat as combined constraints and solve each part consistently.