What is Graphical Solution of Single Linear Inequality? | Simple Explanation for Class 7

S3-SA1-0662

What is the Graphical Solution of a Single Linear Inequality?

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A graphical solution of a single linear inequality is a way to visually represent all the possible values that satisfy the inequality on a graph. Instead of just one point, it shows an entire region on the graph where the inequality holds true.

Simple Example

Quick Example

Imagine you have a budget of Rs. 100 for buying laddoos. If each laddoo costs Rs. 10, the inequality is 10x <= 100 (where x is the number of laddoos). Graphically, this means you can buy 0, 1, 2... up to 10 laddoos, and all these possibilities would be shown as a shaded region on a number line or a plane.

Worked Example

Step-by-Step

Let's find the graphical solution for the inequality x + y > 4.

Step 1: First, treat the inequality as an equation: x + y = 4. This is the boundary line.

---Step 2: Find two points that satisfy the equation. If x = 0, then 0 + y = 4, so y = 4. (Point: (0, 4)). If y = 0, then x + 0 = 4, so x = 4. (Point: (4, 0)).

---Step 3: Plot these two points (0, 4) and (4, 0) on a coordinate plane.

---Step 4: Draw a dashed line connecting these two points. We use a dashed line because the inequality is '>' (greater than), not '>=' (greater than or equal to), meaning points *on* the line are not included in the solution.

---Step 5: Now, choose a test point not on the line. The easiest is usually (0, 0). Substitute (0, 0) into the original inequality: 0 + 0 > 4, which means 0 > 4. This is FALSE.

---Step 6: Since the test point (0, 0) made the inequality false, shade the region on the *opposite* side of the line from (0, 0). If it were true, we would shade the side containing (0, 0).

---Step 7: The shaded region represents all the (x, y) pairs that satisfy x + y > 4. The dashed line indicates that points on the line are not part of the solution.

Answer: The region above and to the right of the dashed line x + y = 4 is the graphical solution.

Why It Matters

Understanding graphical solutions is key for fields like Data Science and AI, where you often need to visualize data boundaries or decision regions. Engineers use this to design systems with limits, and economists use it to model budget constraints or resource allocation. It helps visualize complex problems quickly.

Common Mistakes

MISTAKE: Always drawing a solid line for the boundary. | CORRECTION: Use a solid line for 'less than or equal to' (<=) or 'greater than or equal to' (>=), but use a dashed line for 'less than' (<) or 'greater than' (>).

MISTAKE: Shading the wrong side of the line. | CORRECTION: Always pick a test point (like (0,0) if it's not on the line) and substitute its coordinates into the original inequality. If the inequality is true, shade the side containing the test point. If false, shade the other side.

MISTAKE: Confusing the x and y axes when plotting points. | CORRECTION: Remember that the first number in an ordered pair (x, y) always corresponds to the horizontal (x) axis, and the second number corresponds to the vertical (y) axis.

Practice Questions

Try It Yourself

QUESTION: Graph the inequality y < 2. | ANSWER: Draw a dashed horizontal line at y = 2. Shade the region below this line.

QUESTION: Graph the inequality x >= -1. | ANSWER: Draw a solid vertical line at x = -1. Shade the region to the right of this line.

QUESTION: Graph the inequality 2x + y <= 6. | ANSWER: Draw a solid line connecting (0, 6) and (3, 0). Shade the region containing (0, 0) (below and to the left of the line).

MCQ

Quick Quiz

Which type of line should be used for the inequality x - y < 3?

Solid line

Dashed line

Dotted line

Thick line

The Correct Answer Is:

A dashed line is used for strict inequalities (< or >) because the points on the boundary line itself are not included in the solution set. For <= or >=, a solid line is used.

Real World Connection

In the Real World

Imagine you are a manager at a delivery service like Swiggy or Zomato. You might have an inequality representing the maximum number of orders a rider can handle per hour based on distance and traffic (e.g., total distance covered <= 20 km). Graphing this helps you visualize the 'safe' operating region for your riders, ensuring efficient deliveries without overworking them.

Key Vocabulary

Key Terms

INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, <=, >=, or !=. | LINEAR INEQUALITY: An inequality where the highest power of the variables is 1. | BOUNDARY LINE: The line formed when an inequality is treated as an equation, separating the graph into two regions. | SHADED REGION: The area on the graph that represents all the solutions to the inequality. | TEST POINT: A point chosen not on the boundary line to determine which side of the line to shade.

What's Next

What to Learn Next

Great job learning about single linear inequalities! Next, you can explore 'Systems of Linear Inequalities'. This is where you'll graph two or more inequalities together and find the region that satisfies all of them, which is super useful for more complex real-world problems.