Question: Solve $\dfrac{9}{2}x + 1 < \dfrac{5}{2}x - 2 \le 3x + 1$ and show the answer on a number line.

Step 1. A chain is two inequalities sharing the middle expression. Split it:

$$\text{(i)}\;\; \dfrac{9}{2}x + 1 < \dfrac{5}{2}x - 2 \qquad \text{(ii)}\;\; \dfrac{5}{2}x - 2 \le 3x + 1$$

Step 2. Solve (i):

$$\dfrac{9}{2}x - \dfrac{5}{2}x < -3 \;\Rightarrow\; 2x < -3 \;\Rightarrow\; x < -\dfrac{3}{2}$$

Step 3. Solve (ii):

$$-2 - 1 \le 3x - \dfrac{5}{2}x \;\Rightarrow\; -3 \le \dfrac{1}{2}x \;\Rightarrow\; x \ge -6$$

Step 4. Both must hold — take the overlap:

$$-6 \le x < -\dfrac{3}{2}$$

Number line: filled circle at $-6$ (from $\le$), open circle at $-\frac{3}{2}$ (from $<$), shaded between. If asked for integer values: $-6, -5, -4, -3, -2$.

⚠ Watch out: Never chain-solve across all three parts at once when $x$ lives in every piece — split, solve, intersect. The two circles usually differ (one filled, one open); copying the same style onto both is the giveaway of a rushed answer.

Question: Solve $3x - 5 > 5x + 7$ and represent the answer on a number line.

Step 1. Collect $x$ on one side, numbers on the other — exactly like an equation:

$$3x - 5x > 7 + 5$$ $$-2x > 12$$

Step 2. Divide by $-2$ — and reverse the inequality sign:

$$x < \dfrac{12}{-2} \;\Rightarrow\; x < -6$$

Step 3. Number line: an open circle at $-6$ (the value itself is excluded, since the sign is strict), with the arrow shading everything to the left.

Step 4. Sense-check with one value from the shaded region, say $x = -10$:

$$3(-10) - 5 = -35 \;\; > \;\; 5(-10) + 7 = -43 \; ✓$$

(And $x = -6$ itself gives $-23 > -23$ — false, confirming the open circle.)

Everything else about inequality-solving matches equations exactly (expansion, collecting terms, fractions) - only the negative multiply-or-divide carries the special rule.

⚠ Watch out: The flip happens only when multiplying or dividing by a negative — adding or subtracting anything never flips it. On the number line, $<$ and $>$ take open circles; $\le$ and $\ge$ take filled ones. One test value catches a forgotten flip instantly.