Two-step inequalities work exactly like two-step equations — subtract, then divide (or divide, then subtract). There is one important difference: if you multiply or divide both sides by a negative number, the inequality direction must flip. Positive 4 is greater than positive 2. But multiply both by $-1$ and $-4$ is less than $-2$. The balance tips the other way.
Equations — dividing by negative: safe
$$-2x = 8 \quad \mid \div(-2)$$
$$x = -4$$
The equality stays equal. No flip needed.
Inequalities — dividing by negative: FLIP
$$-2x > 8 \quad \mid \div(-2)$$
$$x < -4$$
The direction reverses. $>$ becomes $<$.
Why does it flip? Think of the number line. $3 > 1$. Multiply both sides by $-1$: $-3$ and $-1$. Now $-3 < -1$ — the order reversed because you reflected both numbers across zero. Multiplying or dividing by any negative number does this reflection.
Remember the rule: Divide or multiply by a negative → flip the sign ($>$ becomes $<$, $\geq$ becomes $\leq$, etc.). Divide or multiply by a positive → no flip needed.
Worked Examples
Worked Example 1 — positive coefficient (no flip)
Mia saves 15 € per week and already has 20 €. She needs more than 110 € for a new saddle. How many weeks does she need to save?
Divide by positive 15 — no flip. Mia needs more than 6 weeks, i.e. at least 7 full weeks.
Worked Example 2 — negative coefficient (flip!)
Copper is on a weight-loss programme. He loses 2 kg per week. His current weight is 510 kg. The vet wants him under 500 kg. After how many weeks will this be achieved?
From here — write the inequality yourself, then solve.
Bedding budget. Mia spends 8 € per week on stable bedding plus a one-off 12 € delivery fee for the first order. She can spend at most 60 € in total. How many weeks can she maintain this?
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Losing weight. Copper currently weighs 502 kg and loses 1.5 kg per week on the new diet. The junior class requires strictly less than 500 kg. After how many weeks does Copper qualify?
Graph:
Score cut-off. Mia starts a dressage test with 100 points. Each mistake costs 4 points. To qualify she needs at least 76 points. What is the maximum number of mistakes she can make?
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Temperature alert. A stable thermometer reads $T$ °C. During the night, the temperature drops 3 °C per hour. The stable must stay above 5 °C for Copper's health. If the current temperature is 20 °C, how many hours until Mia needs to turn on the heating?
Graph:
Challenge. Mia earns $w$ dollars per week. She spends $12 on feed and saves the rest. She needs to save more than $40 per week to meet her goal. Write and solve the inequality. Then write the equivalent inequality using the flip rule to double-check: start from $-(w - 12) < -40$ and solve.
What's next → You've now completed the full equations and inequalities sequence. The next branch of the curriculum is sequences — patterns where each term follows a rule. Arithmetic sequences (add the same amount each time) will look familiar: they're the same patterns you've been writing as equations all along.