Inequalities Masterclass

1. The Symbols

\( < \)
Less than

\( > \)
Greater than

\( \le \)
Less than OR equal to

\( \ge \)
Greater than OR equal to

2. How to Solve Inequalities

We solve inequalities exactly like normal equations (using the balance method), with one major exception.

Standard Method: Treat the inequality sign like an equals sign. Move numbers to one side and \(x\) to the other.

Solve \( 3x + 4 > 19 \)
Step 1: Subtract 4 \( 3x > 15 \)
Step 2: Divide by 3 \( x > 5 \)
(Meaning x can be 6, 7, 5.1, etc., but not 5 or less.)

THE TRAP: Flipping the Sign
If you multiply or divide by a NEGATIVE number, you must FLIP the inequality sign.

Example: \( -2x < 10 \)
Divide by -2: \( x > -5 \)
(Notice the \(<\) changed to \(>\))

3. Number Lines

Visualizing the answer is a common exam question.

Open Circle (○): Use for \( < \) or \( > \) (The number is NOT included).
Closed/Filled Circle (●): Use for \( \le \) or \( \ge \) (The number IS included).

4. Double Inequalities

Example: \( -3 < x \le 4 \)
This means \(x\) is in the "sandwich" between -3 and 4.

List the integers for \( -2 \le x < 1 \)
- It can be -2 (because of the \(\le\)).
- It can be -1, 0.
- It CANNOT be 1 (because of the \(<\)).
Answer: -2, -1, 0

Practice Questions & Solutions

Try these yourself, then click to reveal the solution.

Q1: Solve \( x + 5 < 12 \)

Subtract 5 from both sides.
\( x < 12 - 5 \)
Answer: \( x < 7 \)

Q2: Solve \( 2x \ge 18 \)

Divide both sides by 2.
\( x \ge 9 \)
Answer: \( x \ge 9 \)

Q3: Solve \( 5x - 3 > 17 \)

1. Add 3 to both sides.
\( 5x > 20 \)
2. Divide by 5.
Answer: \( x > 4 \)

Q4: Solve \( 7x + 2 \le 5x + 10 \)

Unknowns on both sides. Move the smallest \(x\) (\(5x\)).
1. Subtract \(5x\) from both sides.
\( 2x + 2 \le 10 \)
2. Subtract 2.
\( 2x \le 8 \)
3. Divide by 2.
Answer: \( x \le 4 \)

Q5: The Trap! Solve \( 10 - x > 4 \)

1. Subtract 10.
\( -x > -6 \)
2. This is really \( -1x > -6 \). We must divide by -1.
FLIP THE SIGN!
Answer: \( x < 6 \)

Q6: Solve \( \frac{x}{3} + 1 < 5 \)

1. Subtract 1.
\( \frac{x}{3} < 4 \)
2. Multiply by 3.
Answer: \( x < 12 \)

Q7: List all integers \(n\) such that \( -1 < n \le 3 \)

Check the left side: \( -1 < n \) means we start at 0 (not -1).
Check the right side: \( n \le 3 \) means we stop at 3 (included).
Answer: 0, 1, 2, 3

Q8: Solve \( 4(x - 2) \ge 12 \)

1. Expand the bracket.
\( 4x - 8 \ge 12 \)
2. Add 8.
\( 4x \ge 20 \)
3. Divide by 4.
Answer: \( x \ge 5 \)

Q9: Solve \( -2x < 8 \)

We need to divide by -2.
Because we divide by a negative, we FLIP the symbol.
\( x > \frac{8}{-2} \)
Answer: \( x > -4 \)

Q10: Solve \( 3 < 2x + 1 \le 11 \)

This is a double inequality. Do the same thing to all three parts.
1. Subtract 1 from everything.
\( 2 < 2x \le 10 \)
2. Divide everything by 2.
\( 1 < x \le 5 \)
Answer: \( 1 < x \le 5 \)