Inequalities are used when we don't have an exact equality, but a relationship like "greater than" or "less than". This lesson covers solving algebraic inequalities and showing the solutions on a number line or as a shaded region on a coordinate plane.

Part A: Solving Linear Inequalities

1. Core Concept

Solving an inequality like 5x - 3 > 7 is almost identical to solving an equation like 5x - 3 = 7. You use inverse operations to isolate the variable. However, there is one critical rule you must never forget.

2. The Golden Rule of Inequalities

When you multiply or divide both sides of an inequality by a NEGATIVE number, you MUST FLIP the inequality sign.

• > becomes <

• < becomes >

• ≥ becomes ≤

• ≤ becomes ≥

Example:

• Solve 10 - 2x ≥ 4

• Subtract 10: -2x ≥ -6

• Divide by -2 and FLIP the sign: x ≤ 3. (The ≥ becomes ≤)

3. Representing Solutions on a Number Line

This is a visual way to show the set of all possible answers.

• Use an open circle (o) for < and > (endpoint is not included).

• Use a closed circle (●) for ≤ and ≥ (endpoint is included).

• Shade the part of the line that represents the solution.

Part B: Representing Inequalities on a Coordinate Plane

This extends the idea to two variables (x, y), where the solution is a whole region.

1. Core Concept

The solution to an inequality like y > 2x + 1 is the set of all coordinate points (x, y) that make the statement true. We show this by shading the correct side of the line.

2. The Method

1. Draw the Boundary Line: First, pretend the inequality is an equation (e.g., for y > x, draw the line y = x).

   • Use a DASHED line for < and >. This shows the line itself is not part of the solution.

   • Use a SOLID line for ≤ and ≥. This shows the line is included in the solution.

2. Shade the Correct Region:

   • For vertical lines (x > a): Shade to the right. For x < a, shade left.

   • For horizontal lines (y > b): Shade above. For y < b, shade below.

   • The Test Point Method (for diagonal lines): If you're not sure which side to shade for y > 2x + 1: a. Pick a simple test point that is NOT on the line, like (0, 0). b. Substitute it into the inequality: 0 > 2(0) + 1, which simplifies to 0 > 1. c. Is this statement true? No, it's false. d. Therefore, shade the side of the line that does NOT contain your test point (0, 0).

4. Exam Tips & Tricks

• "Integer solutions": If a question asks for this, you must list the whole numbers that fit your solution.

  1. Question: "Find the integer solutions for x ≤ 3."

  2. Answer: {..., 0, 1, 2, 3}.

  3. Question: "Find the smallest integer solution for y > -3."

  4. Answer: -2.

• Combining Regions: Exam questions often ask you to shade the region that satisfies multiple inequalities at once (e.g., x > 1, y < 4, and y > x). You do this by:

  1. Drawing all three boundary lines.

  2. Lightly shading the correct side for each one.

The final answer is the region where all the shadings overlap.