Linear Inequalities Class 11 Chapter Notes: Definitions, Rules & Graphs

Key Terminology

Inequality

A mathematical statement that uses symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to) to compare two values or expressions.

Linear Inequality in One Variable

An inequality that can be written in the form ax + b > 0, ax + b < 0, ax + b ≥ 0, or ax + b ≤ 0, where a and b are real numbers and a ≠ 0.

Linear Inequality in Two Variables

An inequality in the form 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.

Solution Set

The set of all values of the variable(s) that make the inequality a true statement. For two variables, this is a region on the Cartesian plane.

Strict Inequality

An inequality that uses the symbols > (greater than) or < (less than). The boundary line is not included in the solution.

Slack Inequality

An inequality that uses the symbols ≥ (greater than or equal to) or ≤ (less than or equal to). The boundary line is included in the solution.

Feasible Region

The common region determined by all the constraints (inequalities) of a system. Any point in this region is a feasible solution.

Rules for Solving Linear Inequalities

• {"point":"Addition/Subtraction: Equal numbers can be added to or subtracted from both sides of an inequality without changing the sign of inequality. If x > y, then x + c > y + c."}
• {"point":"Multiplication/Division by a Positive Number: Both sides of an inequality can be multiplied or divided by the same positive number without changing the sign of inequality. If x < y and c > 0, then xc < yc and x/c < y/c."}
• {"point":"Multiplication/Division by a Negative Number: When both sides are multiplied or divided by a negative number, the sign of inequality is reversed. If x < y and c < 0, then xc > yc and x/c > y/c. This is the most common point of error!"}
• {"point":"Transferring Terms: Any term can be transferred from one side of an inequality to another with its sign changed, without affecting the sign of inequality."}
• {"point":"Graphical Representation (1 Variable): Solutions are represented on a number line. Use an open circle (○) for strict inequalities (<, >) and a closed circle (●) for slack inequalities (≤, ≥)."}
• {"point":"Graphical Representation (2 Variables): The solution is a half-plane. The line itself is dotted for strict inequalities (<, >) and solid for slack inequalities (≤, ≥)."}
• {"point":"Test Point Method: To find the solution region for a two-variable inequality, pick a test point (usually (0,0) if it's not on the line) and check if it satisfies the inequality. If it does, shade the region containing the test point; otherwise, shade the other region."}

Graphical Solution of Linear Inequalities in Two Variables

Graphing is the primary way to visualize the solution set of a linear inequality in two variables. The solution isn't a single point or a line, but an entire region on the Cartesian plane, known as a half-plane. The process involves a few key steps.

First, convert the inequality into an equation. For example, change 2x + 3y > 6 to 2x + 3y = 6. This equation represents the boundary line of your solution region.

Next, draw this line on the graph. A critical distinction must be made here:

• If the inequality is strict (> or <), draw a dotted or dashed line. This signifies that the points on the line are not part of the solution.
• If the inequality is a slack one (≥ or ≤), draw a solid line. This shows that the points on the line are included in the solution set.

Once the boundary line is drawn, it divides the plane into two half-planes. You need to determine which half-plane represents the solution. To do this, pick a convenient test point that is not on the line. The origin, (0, 0), is the easiest choice if the line doesn't pass through it. Substitute the coordinates of the test point into the original inequality.

• If the inequality holds true (e.g., 2(0) + 3(0) > 6 which is 0 > 6, a false statement), then the half-plane containing the test point is not the solution. You must shade the other half-plane.
• If the inequality was, for instance, 2x + 3y < 6, then 0 < 6 would be true, and you would shade the region containing the origin. This shaded region is the graphical representation of the solution set.

How to Solve a System of Linear Inequalities Graphically

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Worked Mini-Examples

• {"header":"Example 1: Solving in one variable","bodyMarkdown":"Solve 3x - 5 < x + 3.\nSolution:\n1. Subtract x from both sides: 2x - 5 < 3\n2. Add 5 to both sides: 2x < 8\n3. Divide by 2 (a positive number, so the sign stays): x < 4\nSolution Set: (-∞, 4)"}
• {"header":"Example 2: Solving by reversing the sign","bodyMarkdown":"Solve 12 - 4x ≥ 20.\nSolution:\n1. Subtract 12 from both sides: -4x ≥ 8\n2. Divide by -4 and reverse the inequality sign: x ≤ -2\nSolution Set: (-∞, -2]"}
• {"header":"Example 3: Identifying the region for a two-variable inequality","bodyMarkdown":"Which region represents x + 2y > 4?\nSolution:\n1. Draw the boundary line x + 2y = 4 as a dotted line (due to >).\n2. Test the origin (0,0): (0) + 2(0) > 4 => 0 > 4, which is false.\n3. Therefore, the solution is the half-plane that does not contain the origin."}

Exam Traps & Scoring Tips

Trap 1: Forgetting to Flip the Inequality Sign. The most frequent error is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. Always double-check this step. For example, -2x < 10 becomes x > -5.

Trap 2: Dotted vs. Solid Lines. Marks are often deducted for using the wrong type of line in graphical solutions. Remember:

• < or > → Dotted Line (boundary not included)
• ≤ or ≥ → Solid Line (boundary included)

Scoring Tip: When solving a system graphically, clearly shade the final feasible region. It's good practice to use different shading patterns or colors for each inequality's solution region and then darken the common area to show the final answer clearly.

Quick Revision Check

• Q: If -5x ≥ 15, what is the correct inequality for x? A: Dividing by -5 reverses the inequality sign, so x ≤ -3.
• Q: Should the graph of 3x - y < 9 have a solid or a dotted line? A: Dotted, because it is a strict inequality (<).
• Q: The test point (0,0) makes the inequality 2x + y > 5 false. Which region do you shade? A: You shade the region that does not contain the origin (0,0).
• Q: What does the solution of a system of linear inequalities represent graphically? A: It represents the common intersection of all the half-planes, known as the feasible region.

Frequently Asked Questions

Frequently Asked Questions