Linear inequality: the sign-flip trap

Like an equation, with ONE trap

Solving an inequality means finding all $x$ that satisfy it.

You can do the same as for an equation:

• Add/subtract the same thing on both sides.
• Multiply/divide by the same positive number.

⚠️ If you multiply or divide by a NEGATIVE number, flip the inequality direction.

Example 1 (no trap)

Solve $3x + 5 < 20$.

$3x < 15$ (subtract: same direction) $x < 5$ (divide by $3 > 0$: same direction)

Solutions: $S = (-\infty, 5)$.

Example 2 (with trap)

Solve $-2x + 3 \geq 11$.

$-2x \geq 8$ (subtract: same direction) $x \leq -4$ (divide by $-2 < 0$: flip direction)

Solutions: $S = (-\infty, -4]$.

Representation on the number line

An inequality has infinitely many solutions. Show them on a number line:

• Open bracket (( )): endpoint excluded (strict $<$, $>$).
• Closed bracket ([ ]): endpoint included (large $\leq$, $\geq$).

Two-sided inequality

Solve $-3 < 2x + 1 \leq 7$.

Subtract $1$ throughout: $-4 < 2x \leq 6$. Divide by $2$ (positive, same direction): $-2 < x \leq 3$.

$S = (-2, 3]$.

Common mistakes

1. Forgetting to flip when multiplying/dividing by a negative.
2. Flipping for no reason (never on addition).
3. Wrong bracket: open vs closed depending on strict/large.
4. Writing $x > 5$ instead of $x < 5$ after flipping everything.

?À toi de jouer

Solve −4x + 8 > 0

A.x < 2B.x > 2C.x < −2D.x > −2

Valider