Linear inequations

An inequation such as $2x - 3 < 7$ is solved the same way an equation is: isolate $x$ by performing the same operation on both sides. Three of the four basic operations behave exactly as they do for equations.

• Adding or subtracting the same quantity from both sides never disturbs the direction of the inequality, whatever that quantity's sign.
• Multiplying or dividing both sides by a positive quantity also leaves the direction unchanged.
• Multiplying or dividing both sides by a negative quantity reverses the inequality sign — this is the one rule that has no equation counterpart, because an equation has no "direction" to flip.

So $-3x < 12$, divided by $-3$, becomes $x > -4$: the $<$ flips to $>$. Nothing else about the arithmetic changes; only the sign symbol turns around. A compound inequation like $-1 \le 2x + 3 < 9$ is really two inequations sharing the middle expression, and both can be processed together by applying the same operation to all three parts at once:

where: the outer numbers $-1$ and $9$ are the given bounds, $2x+3$ is the linear expression being bounded, and the final form $-2 \le x < 3$ names the interval of values $x$ may take — closed (solid) at $-2$ since $\le$ includes it, open (hollow) at $3$ since $<$ excludes it. Once $x$ is isolated, the way you write the answer depends entirely on the number system named in the question, even though the algebra to reach $x \le 5$, say, was identical:

On a number line, $N, W, Z$ are shown as isolated dots (they are discrete sets), while $R$ is shown as continuous shading with an arrow, starting from a solid circle (boundary included) or a hollow circle (boundary excluded).

Question: Solve $\dfrac{7-3x}{2} \ge -1$ for $x \in Z$, and represent the solution on a number line.

Step 1. Multiply both sides by $2$ (positive, so the sign direction is unchanged):

Step 2. Subtract $7$ from both sides:

Step 3. Divide both sides by $-3$. Because $-3$ is negative, reverse the inequality sign:

Step 4. Since $x \in Z$ (integers), the solution is the infinite set of all integers up to and including $3$, written in set-builder form as $\{x : x \le 3,\ x \in Z\}$, i.e. $\{\dots, 0, 1, 2, 3\}$.

Check: substitute $x = 3$ into the original inequation: $\dfrac{7 - 9}{2} = \dfrac{-2}{2} = -1 \ge -1$ ✓, confirming the boundary is correctly included.

The solution is $\boxed{x \le 3,\ x \in Z}$, shown on a number line as solid dots at every integer at $3$ and to its left, with no shading between the dots since $Z$ is a discrete set.

Competitive stretch

For $-2 \le 3x + 1 < 10$, subtract $1$ throughout to get $-3 \le 3x < 9$, then divide by $3$ (positive) throughout to get $-1 \le x < 3$. No sign flip was needed here because $3$ is positive — always check the sign of the multiplier before deciding whether to reverse.

• Forgetting to reverse the sign when multiplying or dividing by a negative number — the single most common error in this topic.
• Reversing the sign anyway when the multiplier is positive, or when only adding/subtracting — reversal applies only to multiplication or division by a negative quantity.
• Mixing up the four number systems: the same simplified inequation $x \le 3$ gives four differently shaped answers depending on whether $x \in N, W, Z,$ or $R$.
• Using a solid (filled) circle for a strict inequality ($<$ or $>$) — strict inequalities need a hollow (open) circle since the boundary value itself is excluded.

Ask the tutor: unsure whether a particular step needs the inequality sign reversed, or how to split a compound inequation like $-2 \le 3x+1 < 10$ — ask the AI tutor to walk through your exact inequation.