What is an inequality?

An inequality is like an equation, but instead of “equals,” it shows a relationship where one side is bigger, smaller, or not equal.

Symbols used in inequalities

Example:

• Means x is any number bigger than 3

How inequalities are different from equations

1. Multiple solutions

   • Equations usually have one solution (x = 2)

   • Inequalities can have many solutions (x > 3 → 3.1, 4, 10, 100…)

2. Graphing on a number line

   • Use open or closed circles:

     • > or < → open circle

     • ≥ or ≤ → closed circle

   • Shade in the direction of solutions

Solving inequalities (step by step)

Inequalities are solved like equations, with one very important rule about multiplying or dividing by negative numbers.

Rule:

If you multiply or divide both sides by a negative number, flip the inequality sign.

Example 1: Simple inequality

Step 1: Subtract 5 from both sides

✅ Solution: all numbers greater than 3

Example 2: Multiply by a positive number

Step 1: Divide both sides by 2

✅ Solution: all numbers less than 5

Example 3: Multiply by a negative number

Step 1: Divide both sides by -3 → flip inequality

✅ Solution: all numbers less than or equal to -3

Graphing inequalities on a number line

• Draw a number line

• Use an open circle for < or >

• Use a closed circle for ≤ or ≥

• Shade left for smaller numbers

• Shade right for bigger numbers

Example:

• Closed circle at 2

• Shade to the right

Graphing inequalities on a coordinate plane

For two variables (x and y), inequalities create a region:

1. Start with the boundary line (replace inequality with =)

2. Use dashed line for < or >

3. Use solid line for ≤ or ≥

4. Shade the side where the inequality is true (test with a point, often (0,0))

Example:

• Dashed line for y = 2x + 1

• Shade below the line

Common beginner mistakes

1. ❌ Forgetting to flip the inequality when multiplying/dividing by negative

2. ❌ Confusing < and > directions

3. ❌ Using open/closed circles incorrectly

4. ❌ Forgetting that inequalities often have many solutions, not just one

Why inequalities matter

• Represent ranges of solutions

• Used in real-world problems: speed limits, budget, distances

• Foundation for linear programming and advanced algebra