Almost identical algebra — add, subtract, multiply, divide to isolate $x$ — with one critical exception: multiplying or dividing both sides by a negative number flips the direction of the inequality sign. Never replace the inequality symbol with "=" partway through your working; that misrepresents the whole answer as a single value instead of a range.

Example: $-3x \le 9$. Dividing both sides by $-3$ gives $x \ge -3$ — notice $\le$ became $\ge$.

Expand first, then gather terms exactly as you would for an equation:

Apply the same operation to all three parts simultaneously:

If dividing by a negative number were needed, both signs in the compound inequality would flip at once.

• Open circle (○): strict inequality ($<$ or $>$) — the boundary value itself is not included.
• Closed/filled circle (●): inclusive inequality ($\le$ or $\ge$) — the boundary value is included.
• An arrow extends from the circle in the direction of all the valid values (right for "greater than," left for "less than").

For the compound solution $-4\le x<4$: a filled circle at $-4$, an open circle at $4$, joined by a solid line between them.

Convert the continuous range into a discrete list by carefully checking each boundary against strict vs. inclusive:

Example: $-2 \le x < 1$. The lowest integer is $-2$ (included), and since $x$ must be strictly less than $1$, the highest integer is $0$. Solutions: $\{-2,-1,0\}$.

Define the variable, translate the constraint directly (e.g. "no more than" becomes $\le$), solve, and sanity-check the answer makes physical sense (a length or cost can't be negative).

Solve $4x - 5 > 15 - x$. State the smallest integer value of $x$ that satisfies the inequality.

1. Add $x$ to both sides: $5x - 5 > 15$.
2. Add 5 to both sides: $5x > 20$.
3. Divide by 5 (positive, so the sign is unchanged): $x > 4$.
4. Since $x$ is strictly greater than 4, the smallest integer solution is $x = 5$.