What this chapter is about
In earlier classes, you learnt to solve equations where two expressions are equal. This chapter extends that idea to situations where one quantity is greater than, less than, or at most equal to another. Such statements are called inequalities, and when the variable appears only to the first power they are linear inequalities.
Linear inequalities arise naturally when resources are limited. For instance, if a student has at most 120 rupees to spend on notebooks and pens, the total cost must be less than or equal to 120; this constraint is an inequality, not an equation. Learning to solve and graph such constraints prepares you for optimisation problems in Class 12 (linear programming) and for reasoning about ranges of values rather than single answers.
After studying this chapter you should be able to solve linear inequalities in one variable algebraically, represent their solutions on a number line, solve systems of two linear inequalities in one variable, and graph linear inequalities in two variables in the coordinate plane. You should also understand how the solution set changes when both sides of an inequality are multiplied or divided by a negative number.
Key ideas
- An inequality uses one of the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to) instead of the equals sign.
- A linear inequality in one variable x has the general form ax + b < 0, ax + b > 0, ax + b ≤ 0, or ax + b ≥ 0, where a ≠ 0.
- Adding or subtracting the same number on both sides of an inequality does not change the direction of the inequality sign.
- Multiplying or dividing both sides by a positive number keeps the inequality sign unchanged; multiplying or dividing by a negative number reverses the inequality sign.
- The solution set of a linear inequality in one variable is usually an infinite interval on the number line, shown with an open circle for strict inequalities and a closed circle when the endpoint is included.
- A linear inequality in two variables, such as 2x + 3y ≤ 12, has a solution set that is a half-plane; the boundary line is included when ≤ or ≥ is used, and excluded when < or > is used.
- A system of linear inequalities in two variables has a solution region that is the intersection of the half-planes corresponding to each inequality.
Formulas and facts to remember
- Rule / Form: If a < b, then a + c < b + c · Meaning: Adding the same number to both sides preserves the inequality.
- Rule / Form: If a < b and c > 0, then ac < bc · Meaning: Multiplying by a positive number preserves direction.
- Rule / Form: If a < b and c < 0, then ac > bc · Meaning: Multiplying by a negative number reverses direction.
- Rule / Form: ax + b > 0 ⇒ x > −b/a (when a > 0) · Meaning: Isolate x; direction stays the same for positive coefficient.
- Rule / Form: ax + b > 0 ⇒ x < −b/a (when a < 0) · Meaning: Direction reverses when coefficient of x is negative.
- Rule / Form: Interval (p, ∞) · Meaning: All real numbers greater than p (p not included).
- Rule / Form: Interval [p, ∞) · Meaning: All real numbers greater than or equal to p (p included).
- Rule / Form: Half-plane for ax + by ≤ c · Meaning: Region on one side of the line ax + by = c, including the line.
Worked examples
Example 1: Solving a linear inequality in one variable
Solve 5x − 7 ≤ 3x + 9 and represent the solution on a number line.
Step 1: Subtract 3x from both sides. 5x − 3x − 7 ≤ 9 2x − 7 ≤ 9
Step 2: Add 7 to both sides. 2x ≤ 16
Step 3: Divide both sides by 2 (positive, so direction unchanged). x ≤ 8
Solution set: all real numbers x such that x ≤ 8, written as (−∞, 8]. On the number line, shade everything to the left of 8 and place a filled circle at 8.
Example 2: Inequality with a negative coefficient
Solve −4x + 5 > 21.
Step 1: Subtract 5 from both sides. −4x > 16
Step 2: Divide both sides by −4. Because we divide by a negative number, reverse the inequality. x < −4
Solution set: (−∞, −4). On the number line, shade to the left of −4 with an open circle at −4.
Example 3: Graphing a linear inequality in two variables
Graph the solution region of x + 2y < 6 in the xy-plane.
Step 1: First draw the boundary line x + 2y = 6. Find two points: When x = 0, y = 3; when y = 0, x = 6. Join (0, 3) and (6, 0) with a dashed line (dashed because the inequality is strict, <).
Step 2: Choose a test point not on the line, say the origin (0, 0). Substitute: 0 + 2(0) = 0, and 0 < 6 is true.
Step 3: Since the origin satisfies the inequality, shade the half-plane containing the origin.
The solution region is the open half-plane below and to the left of the line x + 2y = 6, not including the line itself.
Common mistakes
Forgetting to reverse the inequality when multiplying or dividing by a negative number → always check the sign of the multiplier; if negative, flip the inequality symbol.
Using a solid line for strict inequalities (<, >) when graphing → use a dashed line for < or > and a solid line only for ≤ or ≥.
Testing a point that lies on the boundary line to decide which half-plane to shade → pick any point clearly off the line, such as the origin if it is not on the line.
Writing the solution set incorrectly, for example writing x > 8 when the algebra gives x ≤ 8 → re-check each step, especially when adding or subtracting terms.
Confusing interval notation: writing (8, ∞] instead of (8, ∞) → infinity is never included, so always use a round bracket next to ∞ or −∞.
Quick revision
- Adding or subtracting the same quantity keeps the inequality direction; multiplying or dividing by a negative reverses it.
- Solution of a linear inequality in one variable is an interval on the number line; use open circle for < or >, closed for ≤ or ≥.
- Solution of a linear inequality in two variables is a half-plane; shade the side where a test point satisfies the inequality.
- Dashed boundary for strict inequality, solid boundary when the inequality includes equality.
- Intersection of half-planes gives the solution region for a system of inequalities.