Introducing Linear Inequalities
Solve linear inequalities using the same inverse operations as equations, but with one crucial rule: flip the inequality sign when multiplying or dividing by a negative number. Inequalities describe a range of solutions, not just one value.
You earn $15 per hour and need at least $120 this week. Without calculating formally, write an inequality that represents the number of hours $h$ you need to work.
Before you solve it — what do you think the answer will look like? Will there be one solution or many? How is this different from an equation?
Inequalities are solved using the same inverse operations as equations with one crucial exception: when you multiply or divide both sides by a negative number, you must reverse the inequality sign.
An inequality such as $3x + 5 < 17$ has infinitely many solutions — any $x$ that makes the statement true. The solution is written as a range such as $x < 4$.
Key facts
- An inequality is a statement with $<$, $>$, $\leq$ or $\geq$ instead of $=$.
- Multiplying or dividing by a negative reverses the inequality sign.
- The solution to a linear inequality is a range of values, not a single number.
Concepts
- Why the flip rule is necessary when using negative multipliers.
- The difference between strict ($<$, $>$) and non-strict ($\leq$, $\geq$) inequalities.
- How to verify a solution by substitution.
Skills
- Solve one-step and two-step linear inequalities.
- Solve inequalities involving brackets.
- Check solutions by substituting a test value.
A one-step inequality requires a single inverse operation. A two-step inequality requires two. In both cases, the steps are identical to solving an equation — just carry the inequality sign through.
For example, $x + 8 > 15$: subtract 8 from both sides to get $x > 7$. Every number greater than 7 is a valid solution.
What to write in your book
- Solve inequalities using inverse operations: the same process as equations.
- Write the variable on the left and the solution set on the right: e.g. $x > 7$.
- Check: substitute a value from the solution set into the original inequality.
Bracket inequalities such as $4(x - 1) > 12$ are solved by expanding or dividing first, then applying inverse operations. The flip rule only activates when you multiply or divide by a negative number.
A common error is forgetting to flip when rearranging a negative coefficient inequality such as $-2x \geq 10$. Dividing both sides by $-2$ flips $\geq$ to $\leq$, giving $x \leq -5$.
What to write in your book
- Flip rule: dividing or multiplying by a negative number reverses the inequality sign.
- For bracket inequalities: expand brackets first, then use inverse operations.
- Always show the flip clearly — it is a common source of errors in exams.
After solving, always substitute a test value to verify the solution. Choose a convenient value that should be in the solution set — if the original inequality is satisfied, your answer is likely correct.
In context, state the answer as a sentence: "She needs to work at least 8 hours" rather than just writing $h \geq 8$.
What to write in your book
- Always test a value from your solution set in the original inequality.
- Write the final answer in context if the question is a word problem.
- Check the boundary: test a value just inside and just outside the solution set.
Worked examples · 3 in a row, reveal as you go
Solve $3x + 5 < 17$.
Solve $-2x \geq 10$.
Solve $4(x - 1) > 12$.
What to write in your book
- One-step: single inverse operation. Two-step: undo addition/subtraction first, then multiplication/division.
- Bracket: divide both sides by the coefficient OR expand first.
- Flip rule: only when multiplying/dividing by a negative. Show it explicitly.
- Check: substitute one value inside and one value outside the solution set.
- Solve $x - 3 \geq 8$.
- Solve $5x < 35$.
- Solve $-4x > 20$. Remember the flip rule.
- Solve $2(x + 3) \leq 14$.
The inequality was $15h \geq 120$. Dividing both sides by 15 (positive — no flip): $h \geq 8$. You must work at least 8 hours. The solution set is all real numbers $\geq 8$.
Earlier you guessed the answer might be a range. Confirm this below — why is an inequality a better model than an equation for this type of question?
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Solve $-3x + 6 \leq 18$ and check your solution. (3 marks)
Q2. You earn $15 per hour and need at least $120 this week. Write an inequality and solve it. State your answer in words. (3 marks)
Q3. Explain why the inequality sign reverses when you divide by a negative number. Use a numerical example to support your explanation. (2 marks)
📖 Comprehensive answers (click to reveal)
Practice 1: $x \geq 11$. Practice 2: $x < 7$. Practice 3: $-4x > 20 \Rightarrow x < -5$ (flip). Practice 4: $2(x+3) \leq 14 \Rightarrow x+3 \leq 7 \Rightarrow x \leq 4$.
Q1 (3 marks): $-3x + 6 \leq 18 \Rightarrow -3x \leq 12$ [1]. Divide by $-3$ and flip: $x \geq -4$ [1]. Check: $x = 0$: $-3(0)+6 = 6 \leq 18$ ✓ [1].
Q2 (3 marks): Let $h$ = hours. $15h \geq 120$ [1]. Divide by 15: $h \geq 8$ [1]. She must work at least 8 hours [1].
Q3 (2 marks): Consider $3 > 1$. Multiplying by $-1$ gives $-3 < -1$ — the order reverses [1]. Because inequality preserves order, dividing by a negative must flip the sign to keep the statement true [1].
Solve inequalities at speed, applying the flip rule correctly. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering inequality questions. Pool: lesson 14.
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