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.
Today's hook, You earn $\$15$ per hour at a casual job and need at least $\$120$ this week for a concert ticket. How many hours must you work? This is an inequality problem, and the answer is a range, not just one number.
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Recall, your gut answer first
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Recall, your gut answer first
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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?
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Solving inequalities, the flip rule
Work through the core explanation before applying it.
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Solving inequalities, the flip rule
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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 $x < 4$.
Example: $-2x \geq 10 \Rightarrow x \leq -5$ (sign flips when dividing by $-2$)
Same steps as equations
Add, subtract, multiply, divide, the inverse-operation order is identical to solving an equation. Just keep the inequality sign in place of $=$.
Flip only for negatives
Multiplying or dividing by a positive number does NOT flip the sign. Only negative multipliers/divisors trigger the flip.
Always check your answer
Substitute a value from your solution set into the original inequality to verify it is satisfied.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
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.
Understand
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.
Can do
Skills
Solve one-step and two-step linear inequalities.
Solve inequalities involving brackets.
Check solutions by substituting a test value.
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Key terms
InequalityA mathematical statement using $<$, $>$, $\leq$ or $\geq$ to compare two expressions.
Solution setThe set of all values that make an inequality true, usually an infinite range of numbers.
Flip ruleWhen multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.
Equivalent inequalityAn inequality that has the same solution set as the original but may look different after applying operations to both sides.
Strict inequalityAn inequality using $<$ or $>$ that does NOT include the boundary value in the solution set.
Non-strict inequalityAn inequality using $\leq$ or $\geq$ that DOES include the boundary value in the solution set.
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One-step and two-step inequalities
Work through the core explanation before applying it.
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One-step and two-step inequalities
core concept
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.
Important: When the variable is on the right side, such as $12 \leq 3x$, you can either divide normally or rewrite as $3x \geq 12$ first. Both approaches are valid.
Quick check: which value satisfies $x > 4$?
Inequalities are solved like equations with one extra rule: dividing or multiplying both sides by a negative number flips the inequality sign. One-step: one inverse operation. Two-step: undo addition/subtraction first, then division/multiplication.
Pause, copy the flip rule (dividing or multiplying both sides by a negative number reverses the inequality sign), the two-step solve order for inequalities (undo addition/subtraction first, then multiplication/division), and note that the flip only applies when the operation involves a negative number into your book.
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Bracket inequalities and the flip rule
core concept
We just saw one-step and two-step inequalities solved like equations, with the key extra rule that dividing or multiplying both sides by a negative number flips the inequality sign. That raises a question: what happens when the variable is inside brackets with a positive coefficient outside, does the bracket trigger the flip? This card answers it → a positive coefficient outside brackets never triggers a flip; the flip only occurs when you divide or multiply by a negative number.
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 $x \leq -5$.
Memory tip: Think of the number line, multiplying by a negative reverses order (3 > 1 but −3 < −1), so the sign must flip to preserve the correct relationship.
Which does NOT belong? (Which is not an inequality symbol?)
Bracket inequalities: expand or divide out the coefficient first, then solve. Flip rule: only flips when dividing (or multiplying) by a negative, the bracket’s positive coefficient does not trigger a flip.
Pause, copy the bracket inequality strategies: (1) divide out the positive coefficient first; (2) expand the bracket then solve, and the flip-rule condition (flip only if the bracket's coefficient is negative or you later divide by a negative) into your book.
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Checking and communicating solutions
core concept
We just saw bracket inequalities and the precise condition for the flip, only when dividing or multiplying by a negative number, not simply because brackets are involved. That raises a question: once you have a solution like x > 2, how do you verify it's correct and communicate it in a way that earns full marks? This card answers it → verify by substituting a test value from the solution set into the original inequality; communicate by writing the solution set using inequality notation and checking boundary inclusion (strict or non-strict).
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 $h \geq 8$.
Boundary check: If you solve $3x + 5 < 17$ and get $x < 4$, verify by checking $x = 0$ (in set: $5 < 17$ ✓) and $x = 5$ (outside: $20 < 17$, false, so boundary is correct).
Fill the blank: when dividing both sides of an inequality by $-2$, you must .
After solving an inequality, verify by substituting a test value from the solution set into the original inequality, if the inequality holds, the solution is correct. State the solution set using inequality notation and check boundary inclusion.
Pause, copy the test-value verification method (substitute a value from the solution set into the original inequality; if the inequality holds, the solution is correct) and the boundary-inclusion check (strict > or <: boundary not included; non-strict ≥ or ≤: boundary included) into your book.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · ONE-STEP AND TWO-STEP
Solve $3x + 5 < 17$.
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Subtract 5 from both sides: $3x < 12$
Undo the +5 first (outer operation).
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Divide both sides by 3: $x < 4$
Dividing by positive 3, no flip needed.
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Check: substitute $x = 0$: $3(0) + 5 = 5 < 17$ ✓
Answer: $x < 4$. Any value less than 4 satisfies the inequality.
PROBLEM 2 · FLIP RULE
Solve $-2x \geq 10$.
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Divide both sides by $-2$. Because we are dividing by a negative, flip the sign: $x \leq -5$
All values greater than 4 satisfy the inequality. Note: $x = 4$ does not satisfy it (strict inequality).
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Quick-fire practice
Work through the core explanation before applying it.
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Quick-fire practice
Solve $x - 3 \geq 8$.
Solve $5x < 35$.
Solve $-4x > 20$. Remember the flip rule.
Solve $2(x + 3) \leq 14$.
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Revisit the wage problem
Run the quick drill and copy the summary into your book.
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Revisit the wage problem
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?
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Multiple choice
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Multiple choice
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Short answer
Write full responses, then check them against the model answers.
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Short answer
ApplyBand 43 marks
Q1. Solve $-3x + 6 \leq 18$ and check your solution. (3 marks)
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ApplyBand 43 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)
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UnderstandBand 32 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)
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📖 Comprehensive answers (click to reveal)
Practice 1: $x \geq 11$. Practice 2: $x < 7$. Practice 3: $-4x > 20 \Rightarrow x < -5$ (flip). Practice 4: $2(x+3) \leq 14 \Rightarrow x+3 \leq 7 \Rightarrow x \leq 4$.
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].
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Review and finish
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Boss battle · Inequality Solver
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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.
