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:$2x+1\ge 5$

1. Subtract 1 from both sides: $2x\ge 4$
2. Divide both sides by 2: $x\ge 2$
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 $x\ge -3$ (after dividing by -3 and flipping)

Checking Your Answer

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

For $x\ge 2$: Try $x=5$. If $2\left(5\right)+1\ge 5$, then $11\ge 5$ ✓