Class 11 — Mathematics (NCERT)
Linear Inequalities
Class 11
- ✓By the end of this lesson students will be able to define and identify linear inequalities in one and two variables.
- ✓By the end of this lesson students will be able to solve linear inequalities in one variable algebraically and represent the solution on a number line.
- ✓By the end of this lesson students will be able to solve linear inequalities in two variables graphically.
- ✓By the end of this lesson students will be able to find the solution region for a system of linear inequalities in two variables graphically.
Key concepts
An inequality is a statement involving variables and one of the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations, which show equality, inequalities show a relationship of non-equality between two expressions. \n\nTypes of Inequalities:\n1. Strict Inequalities: Involve < or > (e.g., x < 5, y > -2).\n2. Slack Inequalities: Involve ≤ or ≥ (e.g., x ≤ 5, y ≥ -2).
A linear inequality in one variable is an inequality that can be written in one of the forms ax + b < 0, ax + b > 0, ax + b ≤ 0, or ax + b ≥ 0, where a and b are real numbers and a ≠ 0. The solution to such an inequality is a range of real numbers.\n\nRules for Solving Linear Inequalities:\n1. Adding or subtracting the same number from both sides of an inequality does not change the sign of the inequality.\n Example: If x - 3 > 5, then x - 3 + 3 > 5 + 3, which means x > 8.\n2. Multiplying or dividing both sides of an inequality by a positive number does not change the sign of the inequality.\n Example: If 2x < 10, then 2x/2 < 10/2, which means x < 5.\n3. Multiplying or dividing both sides of an inequality by a negative number reverses the sign of the inequality.\n Example: If -3x ≥ 9, then -3x/(-3) ≤ 9/(-3), which means x ≤ -3.\n\nRepresentation on a Number Line:\n- For strict inequalities (< or >), use an open circle at the boundary point to indicate that the point is not included in the solution set.\n- For slack inequalities (≤ or ≥), use a closed (filled) circle at the boundary point to indicate that the point is included in the solution set.\n- Shade the portion of the number line that represents the solution.
A linear inequality in two variables is an inequality that can be written in one of the forms ax + by < c, ax + by > c, ax + by ≤ c, or ax + by ≥ c, where a, b, and c are real numbers, and a and b are not both zero. The solution to such an inequality is a region in the Cartesian plane.\n\nGraphical Solution Method:\n1. Replace the inequality sign with an equality sign to obtain the equation of the boundary line (ax + by = c).\n2. Plot this line on the Cartesian plane.\n - If the inequality is strict (< or >), draw a dashed or broken line to indicate that points on the line are not part of the solution.\n - If the inequality is slack (≤ or ≥), draw a solid line to indicate that points on the line are part of the solution.\n3. Choose a test point (a point not on the boundary line). The origin (0,0) is often a convenient test point, provided it does not lie on the line.\n4. Substitute the coordinates of the test point into the original inequality.\n - If the test point satisfies the inequality, then the region containing the test point is the solution region. Shade this region.\n - If the test point does not satisfy the inequality, then the region opposite to the test point is the solution region. Shade this region.
A system of linear inequalities consists of two or more linear inequalities involving the same variables. The solution to a system of linear inequalities is the region in the Cartesian plane that satisfies all the inequalities simultaneously. This region is the intersection of the solution regions of individual inequalities.\n\nGraphical Solution Method:\n1. Graph each inequality separately on the same Cartesian plane using the method described for a single linear inequality in two variables.\n2. The common region (intersection) that is shaded by all the inequalities represents the solution set for the system of inequalities. This region is called the feasible region.
Key facts to remember
- 1When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.
- 2For strict inequalities (< or >), the boundary line is dashed, indicating points on the line are not part of the solution.
- 3For slack inequalities (≤ or ≥), the boundary line is solid, indicating points on the line are part of the solution.
- 4The origin (0,0) is a convenient test point for determining the solution region, provided it does not lie on the boundary line.
- 5The solution to a system of linear inequalities is the intersection (common region) of the solution sets of all individual inequalities.
- 6Solutions to one-variable inequalities are represented on a number line, while solutions to two-variable inequalities are regions in the Cartesian plane.
Worked examples
Example 1
Solve 3(x - 1) ≤ 2(x - 3) + 5 for real x and represent the solution on a number line.
Answer
The solution is x ≤ 2. On a number line, this is represented by a closed circle at 2 and shading to the left.
Remember to simplify expressions on both sides before isolating the variable.
Example 2
Solve 2x + 3y > 6 graphically.
Answer
The solution is the half-plane above the dashed line 2x + 3y = 6, not including the line itself.
Always use a ruler and pencil for drawing graphs. Clearly label the axes and the boundary line.
Example 3
Solve the following system of linear inequalities graphically: \n1. x + y ≤ 5 \n2. x - y < 3
Answer
The solution is the feasible region (common shaded area) bounded by the solid line x + y = 5 and the dashed line x - y = 3, including the points on x + y = 5 but not on x - y = 3.
Use different colours or shading patterns for each inequality's solution to clearly identify the common region.
Common mistakes
- ✗Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
- ✗Incorrectly drawing the boundary line as solid instead of dashed (or vice-versa) for strict/slack inequalities.
- ✗Shading the incorrect half-plane after using a test point.
- ✗Not identifying the correct common region for a system of inequalities, especially when lines intersect.
- ✗Making calculation errors while simplifying expressions or finding points for the boundary line.
Exam tips
- ★Always show all steps clearly, especially when solving algebraically, as partial credit may be awarded.
- ★For graphical solutions, use a sharp pencil and a ruler. Label the axes (X, Y), the origin (O), and the equations of the boundary lines clearly.
- ★When solving a system of inequalities, use distinct shading patterns or colours for each inequality's solution to make the common feasible region easily identifiable.
- ★Double-check your solution by picking a point from your shaded region and substituting it into the original inequality/inequalities to ensure it satisfies them.
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