Solving an inequality and the sign trap

The basic rules

An inequality is solved in the same way as an equation, by isolating x, but there are two operations that require particular care:

1. You can add or subtract the same number from both sides without changing the direction of the inequality.
2. You can multiply or divide both sides by a strictly positive number without changing the direction of the inequality.
3. If you multiply or divide by a strictly negative number, you must reverse the direction of the inequality.

Step-by-step example

Let’s solve 2x + 3 < 9:

• 2x + 3 < 9
• 2x < 9 - 3 (we subtract 3)
• 2x < 6
• x < 3 (we divide by 2, which is positive, so the direction does not change)

The set of solutions is: x < 3, i.e. the interval ]∞; 3[.

The pitfall of negative numbers

Let’s solve -3x + 6 > 0:

• -3x + 6 > 0
• -3x > -6 (we subtract 6)
• x < 2 (we divide by -3; as this is negative, we REVERSE the direction: > becomes <)

If we forget to reverse the direction, we get x > 2, which is false! Checking with x = 0: -3 × 0 + 6 = 6 > 0 is true, and 0 < 2 is consistent with the correct solution x < 2.

Representation on a number line

For x < 3: we mark an empty circle (or an open bracket) at 3, with an arrow pointing to the left (minus infinity). For x >= -1, we mark a solid dot (or a closed bracket) at -1, with an arrow pointing to the right.