Solving Linear Inequalities: Lesson, Examples and Practice

1Learning goal: what you will be able to do

An equation says two things are equal. An inequality says one is bigger or smaller, and its answer is a whole range of values, not a single number. By the end of this lesson you will be able to solve a linear inequality, show the answer on a number line, and know exactly when the inequality sign must be reversed.

Time45 to 55 minutes

You needPen, paper and a calculator

AssumedSolving linear equations (Lesson 1 of this module), negative numbers

Learning outcomes

FoundationRead and write the symbols <, ≤, > and ≥, and show a solution set on a number line with open and closed circles.
CoreSolve one-step and two-step linear inequalities, and reverse the sign when multiplying or dividing by a negative number.
CoreSolve inequalities with the unknown on both sides and double inequalities, and list the integers that satisfy them.
ExtensionForm an inequality from a word problem, and recognise inequalities with no solution or with every number as a solution.

Where this lesson fits

This is the second lesson of Module 3 in Algebra Foundations. It builds directly on solving equations: the steps are the same, with one important new rule. The lesson after it, on straight-line graphs, is planned and not yet published.

Module 3, Lesson 1: Solving linear equationsOpen
Module 3, Lesson 2: Linear inequalitiesThis lesson
Module 3, Lesson 3: Straight-line graphs and equationsOpen
Module 4, Simultaneous equationsPlanned

Where this topic appears in courses

Linear inequalities are taught in lower and upper secondary school: UK Key Stage 3 and GCSE Mathematics, Cambridge IGCSE Mathematics, WAEC senior secondary mathematics, IB Middle Years and Diploma mathematics, and US Common Core standards such as 7.EE.B.4 and HSA-REI.B.3. Courses differ in how far they go, for example whether they include double inequalities or graphing in two variables. This lesson supports, and does not replace, the specification you are following.

2A familiar example: the taxi again

In the last lesson a taxi charged a fixed $3 plus $2 per kilometre, and you had exactly $17. This time you have up to $25. How far can you go?

Predict first: what is the greatest number of whole kilometres you can travel?

Work it out as you did before, but now ask: what is the most I can spend?

11 km. Take away the $3 fixed charge, leaving $22. Divide by $2: 11. At 11 km you pay $3 + $22 = $25 exactly. At 12 km you would pay $27, which is too much. Any distance up to 11 km works, so the answer is a range, not a single value.

The cost must be at most $25. Let k be the distance in kilometres:

3 + 2k ≤ 25

Solving it follows the same reverse-the-operations idea as an equation: subtract 3, then divide by 2, giving k ≤ 11. The only new ideas are what the symbol means, how to draw the whole range, and one rule that surprises many students. Explore with the tool below.

Try it: the inequality explorer

Build an inequality of the form ax + b (sign) c. The tool solves it and draws the solution. Choose a negative a and watch what happens to the sign. Then type a test number to check whether it belongs to the solution set.

Sign

Test a number

A filled circle means the boundary value is included (≤ or ≥). An open circle means it is not (< or >). The arrow shows the direction in which the solutions continue forever.

3Clear explanation

The four symbols

Symbol	Read as	Example	On a number line
<	is less than	x < 3	Open circle at 3, arrow left
≤	is less than or equal to	x ≤ 3	Filled circle at 3, arrow left
>	is greater than	x > 3	Open circle at 3, arrow right
≥	is greater than or equal to	x ≥ 3	Filled circle at 3, arrow right

A solution of an inequality is any value that makes it true. Unlike an equation, an inequality usually has infinitely many solutions, which is why we describe them as a solution set. The point where the inequality changes from true to false is the boundary value.

Same steps as equations

You may add or subtract the same number on both sides, and you may multiply or divide both sides by the same positive number. The sign stays exactly as it was. For example, 3x − 4 > 11 becomes 3x > 15, then x > 5.

The one new rule: reverse the sign for a negative

When you multiply or divide both sides by a negative number, the inequality sign must be reversed. Here is why. Start with a true statement, 2 < 5. Multiply both sides by −1. You get −2 and −5. On the number line −2 is to the right of −5, so −2 > −5. Multiplying by a negative flips the order, so the sign must flip to keep the statement true.

Memory check: the flip happens only when you multiply or divide by a negative number. Adding or subtracting a negative number does not flip anything. And the boundary value does not change; only the direction does.

An alternative that avoids the rule: move the x term to the side where it is positive. For −2x + 3 > 11, add 2x to both sides to get 3 > 11 + 2x, then 2x < −8, so x < −4. Both routes give the same answer.

Unknown on both sides and double inequalities

Collect the x terms on one side exactly as you did for equations. A double inequality such as −3 < 2x + 1 ≤ 9 means both conditions at once. Apply every operation to all three parts, then read the result as a range. To list integers, count whole numbers inside the range, and check carefully whether each end is open or closed.

Checking an answer

Pick one number inside your solution set and one outside it, and substitute both into the original inequality. The inside number must give a true statement and the outside number a false one. Also test the boundary value itself: it must work for ≤ and ≥ and fail for < and >.

Forming an inequality from words

Words	Symbol
at most, no more than, up to, maximum	≤
at least, no fewer than, minimum	≥
less than, fewer than, below	<
more than, greater than, above	>

Common mistakes

Mistake	Why it is wrong	Fix
Forgetting to reverse the sign when dividing by a negative	The order of numbers reverses on a negative scale	Circle the negative before you divide, and flip
Reversing the sign when subtracting a number	Only multiplying or dividing by a negative flips it	Flip only for × or ÷ by a negative
Using a filled circle for < or >	The boundary value does not satisfy a strict inequality	Open for strict, filled for ≤ and ≥
Giving a single number as the answer	An inequality has a range of solutions	Write x < 4, not x = 4
Counting integers without checking the ends	Whether an end is included changes the count	List the integers and test the end values

4Worked examples

Read each example with a pencil in hand. Cover the steps, try the first line yourself, then compare. Each ends with a check using one number inside and one outside the solution set.

Example 1 Foundation 2 marks: a one-step inequality

Solve x + 7 > 12 and describe the number line.

Subtract 7 from both sides: x > 5.
Number line: an open circle at 5 (5 itself is not included) with an arrow pointing right.
Check: x = 6 gives 13 > 12, true. x = 5 gives 12 > 12, false. x = 4 gives 11 > 12, false.

Answer: x > 5.

Example 2 Core 3 marks: a two-step inequality

Solve 3x − 5 ≤ 16.

Add 5 to both sides: 3x ≤ 21.
Divide both sides by 3, a positive number, so the sign stays: x ≤ 7.
Number line: a filled circle at 7 with an arrow pointing left.
Check: x = 7 gives 16 ≤ 16, true. x = 8 gives 19 ≤ 16, false. x = 0 gives −5 ≤ 16, true.

Answer: x ≤ 7.

Example 3 Core 3 marks: dividing by a negative

Solve −2x + 3 > 11.

Subtract 3 from both sides: −2x > 8.
Divide both sides by −2. This is a negative number, so reverse the sign: x < −4.
Number line: an open circle at −4 with an arrow pointing left.
Check: x = −5 gives −2(−5) + 3 = 13 > 11, true. x = −4 gives 11 > 11, false. x = 0 gives 3 > 11, false.

Answer: x < −4.

Example 4 Core 4 marks: the unknown on both sides

Solve 5x − 3 < 2x + 9.

Subtract 2x from both sides: 3x − 3 < 9.
Add 3: 3x < 12.
Divide by 3: x < 4.
Check: x = 3 gives 12 < 15, true. x = 4 gives 17 < 17, false.

Answer: x < 4.

Example 5 Core 4 marks: a double inequality and integers

Solve −3 < 2x + 1 ≤ 9 and list the integer solutions.

Subtract 1 from all three parts: −4 < 2x ≤ 8.
Divide all three parts by 2: −2 < x ≤ 4.
The ends differ: −2 is excluded (strict) and 4 is included.
Integers: −1, 0, 1, 2, 3, 4. That is 6 integers.
Check the ends: x = 4 gives 2(4) + 1 = 9 ≤ 9, true. x = −2 gives −3, and −3 < −3 is false.

Answer: −2 < x ≤ 4, and the integers are −1, 0, 1, 2, 3, 4.

Example 6 Core 4 marks: a word problem

A phone plan costs $15 a month plus $0.05 per minute of calls. You can spend at most $40 a month. What is the greatest number of call minutes?

Define. Let m be the number of minutes.
Translate. "At most $40" means ≤: 15 + 0.05m ≤ 40.
Solve. Subtract 15: 0.05m ≤ 25. Divide by 0.05: m ≤ 500.
Check. At 500 minutes: 15 + 0.05 × 500 = 15 + 25 = $40, which is allowed.

Answer: at most 500 minutes.

Example 7 Extension 4 marks: no solution or every number

(a) Solve 2(x + 1) > 2x + 5. (b) Solve 3x + 1 < 3x + 2.

(a) Expand: 2x + 2 > 2x + 5. Subtract 2x from both sides: 2 > 5. This is false for every x, so there is no solution.
(b) Subtract 3x from both sides: 1 < 2. This is true for every x, so every real number is a solution.

Answer: (a) no solution; (b) all real numbers.

6Answer explanations

Attempt every question before you open these. A right answer with a wrong method is luck, so read each explanation even when you were correct. Questions 3 and 6 show their own worked solution as soon as you answer.

Question 1: when to reverse the sign

Answer: when both sides are multiplied or divided by a negative number. A negative scale reverses the order of numbers, so the sign must reverse to keep the statement true.

Why the others fail: subtracting never reverses the sign (a). A negative answer does not matter, only the operation does (b). An x on both sides is handled by subtracting, which does not flip anything (d).

Question 2: 3x − 7 ≥ 11

Add 7: 3x ≥ 18. Divide by 3 (positive, so the sign stays): x ≥ 6. Check: 3(6) − 7 = 11, and 11 ≥ 11 is true.

Question 3: generated inequality

Undo the addition or subtraction, then divide. If the coefficient of x is negative, the direction of the sign reverses, but the boundary value does not change. Use New numbers to practise both the positive and the negative case.

Question 4: reading a number line

Answer: x ≥ 2. A filled circle means 2 is included, and the arrow to the right means all larger values are included. An open circle would mean x > 2.

Question 5: −2x + 5 < 13

Subtract 5: −2x < 8. Divide by −2 and reverse the sign: x > −4.

Check: x = 0 gives 5 < 13, true, and 0 > −4. x = −6 gives 17 < 13, false, and −6 is not greater than −4. Forgetting to reverse would give x < −4, which fails this check.

Question 6: generated integer count

Solve the double inequality by applying each step to all three parts, then list the whole numbers in the range. The commonest slip is miscounting an end: strict signs exclude the boundary, and ≤ includes it. A quick check is to substitute the first and last integer you listed.

Question 7: the student's error

Answer: x < −4. Dividing both sides of −3x > 12 by −3 gives x < −4, because the sign reverses. The student divided correctly but did not flip.

Check: x = −5 gives −3(−5) = 15 > 12, true. x = 0 gives 0 > 12, false.

Question 8: the taxi with $25

3 + 2k ≤ 25, so 2k ≤ 22 and k ≤ 11. The greatest whole number is 11. At 11 km the fare is exactly $25, which is allowed.

Question 9: 5n − 12 > 30

Add 12: 5n > 42. Divide by 5: n > 8.4. The smallest integer greater than 8.4 is 9.

Check: n = 9 gives 33 > 30, true. n = 8 gives 28 > 30, false. Answering 8 comes from rounding 8.4 to the nearest integer; with a strict > you must go up to the next integer above the boundary.

7Short recap

An inequality compares two expressions. Its solution is a range of values, shown on a number line.
Open circle for < and >. Filled circle for ≤ and ≥. The arrow shows the direction.
Add, subtract, multiply or divide by a positive number: the sign stays.
Multiply or divide by a negative number: reverse the sign. The boundary value does not change.
Collect x terms on one side exactly as in equations.
For a double inequality, do every step to all three parts, then read off the range and count integers carefully.
Check with one number inside, one outside, and the boundary itself.
"At most" means ≤, "at least" means ≥. If the x terms cancel, a false leftover means no solution and a true leftover means every number.

Tick what you can now do

I can read the four symbols and show a solution on a number line.
I can solve one-step and two-step inequalities, reversing the sign when needed.
I can solve with x on both sides and double inequalities and list the integers.
I can form an inequality from a word problem and spot no-solution cases.

If you ticked fewer than four, return to the matching worked example, redo it without looking, then try the generated questions again. The next lesson uses these ideas to read straight-line graphs.

Related Encyclopedia entries

Algebra: the wider branch of mathematics this lesson belongs to.
Solving Linear Equations and Straight-Line Graphs and Equations: the other lessons in this module.

References and further reading

Titles and publishers only. Search each site for the title to reach the exact page.

OpenStax, Elementary Algebra 2e, the chapter on solving linear equations and inequalities. Free and open textbook. https://openstax.org/
Khan Academy, Algebra 1, unit on solving equations and inequalities. https://www.khanacademy.org/
National Governors Association Center for Best Practices and Council of Chief State School Officers, Common Core State Standards for Mathematics. https://www.thecorestandards.org/
UK Department for Education, Mathematics programmes of study: key stage 3. https://www.gov.uk/