How to Solve Linear Inequalities with Examples and Graphs
The concept of linear inequalities plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Linear Inequalities?
A linear inequality is an algebraic statement that compares two linear expressions using the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). For example, \( 3x + 2 < 7 \) is a linear inequality in one variable. You’ll find this concept applied in areas such as algebraic problem-solving, graphical analysis, and real-world word problems involving limits or constraints.
Key Symbols and Formula for Linear Inequalities
Here are the main symbols used in linear inequalities:
| Symbol | Meaning | Example |
|---|---|---|
| < | Less Than | x < 2 |
| > | Greater Than | x > -3 |
| ≤ | Less Than or Equal To | 3x ≤ 9 |
| ≥ | Greater Than or Equal To | y ≥ 0 |
Here’s the standard linear form: \( ax + b \; \# \; c \), where \( \# \) can be <, >, ≤, or ≥.
Types of Linear Inequalities
Linear inequalities are mainly of these types:
- In one variable (e.g., \( 2x - 5 < 11 \))
- In two variables (e.g., \( x + y \geq 6 \))
- System of linear inequalities (e.g., a set of two or more inequalities like \( x + y > 4 \), \( x - y \leq 2 \))
Step-by-Step Illustration
Let's learn how to solve a simple linear inequality in one variable:
1. Start with the given: \( 2x + 3 > 7 \ )2. Subtract 3 from both sides: \( 2x > 4 \)
3. Divide both sides by 2: \( x > 2 \)
4. Final Answer: All values of x greater than 2 are solutions.
Rules to Remember for Solving Linear Inequalities
- If you multiply or divide both sides by a negative number, reverse the inequality sign.
Example: \( -2x < 6 \Rightarrow x > -3 \) - Addition or subtraction of the same number on both sides does NOT change the sign.
Graphical Representation of Linear Inequalities
Solutions to linear inequalities can be shown visually:
- For one variable: Draw a number line, use open or closed dots depending on whether the boundary is excluded or included.
- For two variables: Draw a straight boundary line on the xy-plane. Use shading to indicate the solution region. Dotted boundary for < or >, solid boundary for ≤ or ≥.
Try These Yourself
- Solve: \( -3x + 5 \leq 2 \ )
- Graph: \( y > 2x - 1 \ ) on the coordinate plane.
- Check if \( x = 4 \) satisfies \( 2x - 3 < 7 \).
- List three real-life problems that use linear inequalities.
Frequent Errors and Misunderstandings
- Forgetting to flip the inequality sign when multiplying or dividing by a negative.
- Treating linear inequalities just like equations (remember: the solution set is usually infinite and shown as ranges or regions).
- Plotting wrong boundary dots (open/closed confusion) on the number line or graph.
Linear Inequalities in Word Problems
Linear inequalities are used in scenarios like budgeting, comparing marks, age restrictions, or minimum/maximum constraints. For example, if a train ticket costs at least ₹50: \( x \geq 50 \). In business, inequalities help model profit and loss limits.
Speed Trick or Vedic Shortcut
When you solve linear inequalities in an exam, first move all variables to one side and constants to the other. Always double-check multiplication/division steps, especially with negatives, and quickly test a boundary value to confirm your solution.
Example Trick: For two-variable inequalities: substitute (0,0) into the inequality after graphing to decide which region to shade (this is called the “test point” method).
Tricks like these are taught in Vedantu sessions to help students build accuracy and confidence.
Relation to Other Concepts
The idea of linear inequalities connects closely with linear equations and systems of equations. Mastering inequalities also helps with graphing skills and later on with linear programming.
Classroom Tip
A simple way to remember: If you multiply/divide both sides by a negative number, flip the sign (< becomes >, and vice versa). Vedantu’s teachers always remind students with mnemonics like “Negative = Flip Sign!” during live doubt-clearing classes.
Wrapping It All Up
We explored linear inequalities—from definition, symbols, formulas, graphs, stepwise examples, and related concepts. Continue practicing with Vedantu’s online resources and free study notes to become confident at solving all types of linear inequalities quickly and accurately.
Must-Visit Links for Further Revision
FAQs on Linear Inequalities: Concepts, Solving Methods & Graphs
1. What is a linear inequality in Maths as per the Class 11 syllabus?
A linear inequality is a mathematical statement that involves a linear function and an inequality symbol (<, >, ≤, or ≥). Unlike a linear equation which equates two expressions and typically has a single solution, a linear inequality compares them, resulting in a range of possible values as the solution. For instance, 2x + 3 < 9 is a linear inequality in one variable.
2. What is the main difference between solving a linear equation and a linear inequality?
The primary difference lies in the solution and one critical operation. A linear equation's solution is a specific value (e.g., x = 3). A linear inequality's solution is a set of values or an interval (e.g., x > 3). The critical operational difference is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign to maintain the truth of the statement.
3. What are the fundamental rules for solving linear inequalities?
The fundamental rules for solving linear inequalities are similar to those for equations, with one key exception:
- You can add or subtract the same number on both sides without changing the inequality sign.
- You can multiply or divide both sides by the same positive number without changing the inequality sign.
- If you multiply or divide both sides by a negative number, the inequality sign must be reversed (e.g., < becomes >).
4. How does graphing a linear inequality in one variable differ from graphing one in two variables?
Graphing these two types of inequalities differs significantly in terms of dimensions and representation:
- One-Variable Inequality (e.g., x ≤ 2): This is graphed on a number line. The solution is represented by a shaded ray or line segment. A solid dot is used for ≤ or ≥, while an open circle is used for < or >.
- Two-Variable Inequality (e.g., y > 2x - 1): This is graphed on a Cartesian (x-y) plane. The solution is an entire region, or a half-plane, on one side of a boundary line. The region is shaded to indicate all coordinate pairs (x, y) that satisfy the inequality.
5. How do you decide whether to use a solid or a dashed line when graphing a two-variable inequality?
The choice between a solid and a dashed line depends on whether the inequality is strict or inclusive:
- Use a solid line for inequalities with ≤ (less than or equal to) or ≥ (greater than or equal to). This indicates that the points on the line itself are part of the solution set.
- Use a dashed line for strict inequalities with < (less than) or > (greater than). This shows that the points on the line are not included in the solution set.
6. How does the 'test point' method help determine the solution region for a two-variable inequality?
The test point method is a simple way to identify which half-plane to shade. After graphing the boundary line, you select a point not on the line (the origin (0,0) is easiest if the line doesn't pass through it). You substitute the coordinates of this test point into the original inequality. If the resulting statement is true, you shade the entire region containing the test point. If it is false, you shade the other region.
7. Why must the inequality sign be reversed when multiplying or dividing by a negative number?
Reversing the sign is necessary to preserve the mathematical truth of the inequality. Multiplying or dividing by a negative number essentially 'flips' the numbers' positions relative to zero on the number line. For example, we know that 2 < 5. If we multiply both sides by -1 without reversing the sign, we get -2 < -5, which is false. To make it true, we must reverse the sign to get -2 > -5.
8. What happens if the variable is eliminated while solving a linear inequality?
If the variable terms cancel out completely during the solving process, you are left with a statement comparing two numbers. The solution depends on this final statement:
- If the statement is true (e.g., 3 < 7), the original inequality is true for all real numbers. The solution is all real numbers, denoted as (-∞, ∞).
- If the statement is false (e.g., 3 > 7), the original inequality has no solution.
9. What is a 'feasible region' in the context of a system of linear inequalities?
A feasible region is the graphical solution to a system of two or more linear inequalities. It represents the set of all points (x, y) that simultaneously satisfy all the inequalities in the system. This region is found by graphing all the individual inequalities on the same coordinate plane and identifying the area where all the shaded regions overlap. This concept is foundational to linear programming.
10. What are some real-world examples where linear inequalities are applied?
Linear inequalities are widely used to model real-world situations involving constraints and optimization. Examples include:
- Budgeting: Determining spending limits, such as 'the cost of notebooks and pens must be less than or equal to ₹500'.
- Business Production: A company planning to produce at least 100 widgets per day, subject to constraints on labour and materials.
- Nutrition Planning: Creating a diet plan where caloric intake must be below a certain limit and vitamin intake must be above a minimum requirement.
- Time Management: Allocating study hours for different subjects, where the total time cannot exceed the available hours.
