Year 7: Solve Inequalities in a Single Variable

What is an Inequality?

An inequality is like an equation, but instead of using "=" we use comparison symbols:

• > means "greater than"
• < means "less than"
• >= means "greater than or equal to"
• <= means "less than or equal to"

Steps to Solve Inequalities

1. Treat it like an equation: Use the same operations as for equations
2. Add or subtract: Move constants just like in equations
3. Multiply or divide: Do the same to both sides
4. FLIP the symbol: If you multiply or divide by a negative number, flip the inequality sign
5. Check your solution: Test a value in your range to verify

Examples

Example 1: 2x + 1 >= 5

1. Subtract 1 from both sides: 2x >= 4
2. Divide both sides by 2: x >= 2
3. This means x can be any number greater than or equal to 2

Example 2: x - 3 < 5

1. Add 3 to both sides: x < 8
2. This means x can be any number less than 8

Example 3: -2x > 8 (IMPORTANT: Notice the negative coefficient)

1. Divide both sides by -2: x < -4 (FLIP THE SYMBOL!)
2. This means x can be any number less than -4

Example 4: 3x - 2 <= 10

1. Add 2 to both sides: 3x <= 12
2. Divide both sides by 3: x <= 4
3. This means x can be any number less than or equal to 4

The Critical Rule: Flipping the Symbol

When you multiply or divide an inequality by a NEGATIVE number, you must flip the inequality symbol:

• -x > 5 becomes x < -5 (after dividing by -1 and flipping)
• -3x <= 9 becomes x >= -3 (after dividing by -3 and flipping)

Checking Your Answer

Pick a value in your solution range and check it works:

For x >= 2: Try x = 5. If 2(5) + 1 >= 5, then 11 >= 5 ✓