Is your child struggling with linear inequalities in Secondary Math? You’re not alone. Many students find inequalities confusing at first, especially when it comes to knowing when to flip that inequality sign! But here’s the good news: once you understand the basic rules, solving linear inequalities becomes just as straightforward as solving equations.
The truth is, your child already uses inequalities every day without realising it. When they say “I need at least $15 for lunch,” or “The exam is in less than 3 weeks,” they’re thinking in terms of inequalities! Understanding this connection makes the math feel less abstract and more practical.
In this guide, we’ll break down everything you need to know about linear inequalities, from real-world examples to step-by-step solving methods. Whether your child is in Sec 1 or Sec 2, this article will help them master this important math topic with confidence.
Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
What Are Linear Inequalities?
A linear inequality is similar to a linear equation, but instead of an equal sign (=), it uses inequality symbols like:
- > (greater than)
- < (less than)
- ≥ (greater than or equal to)
- ≤ (less than or equal to)
For example:
- Equation: 2x + 3 = 7
- Inequality: 2x + 3 > 7
While equations have one specific solution, inequalities have a range of solutions. This means your child isn’t looking for just one number—they’re finding all the values that make the inequality true.
Real-life Examples of Inequalities
Before we dive into solving techniques, let’s help your child see where inequalities appear in everyday life. This makes the topic feel relevant and easier to grasp!
Height restrictions
A ride at an amusement park may require you to be at least 140 cm tall. This means the minimum acceptable height, h, is 140 cm and anything above it is acceptable. In inequality form: h ≥ 140.
Money and spending
A shop may offer a gift only if you spend at least 20 dollars. That means your spending amount, x, must satisfy x ≥ 20. On the other hand, if your budget for a school trip is no more than 50 dollars, then x ≤ 50.
Capacity and safety limits
A lift has a maximum safe load of 900 kg. This means the total load, w, must not exceed an upper limit. The inequality is w ≤ 900.
Timing and performance
A student may need to complete a run in less than 15 minutes to earn a fitness award. The time taken, t, must satisfy t < 15.
Science and experiment control
A chemical reaction may need the temperature to stay between 25 and 40 degree Celsius inclusive. This forms a compound inequality: 25 ≤ T ≤ 40.
Daily routines
Each day, you spend h hours on gaming and r hours on revision. As a personal goal to spend at least 3 hours on revision daily and spend at most 2 hours on gaming daily, then you would have : h ≤ 2 and r ≥ 3.
The examples above show that inequalities describe situations with flexibility. There is never just one correct answer; instead there is a whole set of values that meet the condition(s). This is different from equations, where we search for specific unknown values.
Linear inequalities specifically involve expressions where the variable appears with power 1. Examples are x + 3 > 7 or 2x − 5 ≤ 11. These are the types of inequality you will encounter most frequently at lower secondary level. They form the foundation for more advanced work in algebra, linear graphs, real-world modelling, and topics such as quadratic inequalities in upper secondary.
*Quadratic Inequalities is taught in Additional Mathematics. You can download a free copy of our Additional Mathematics revision notes here.
Understanding linear inequalities is therefore an essential skill. Once you learn how to interpret the comparison symbols, translate everyday conditions into inequalities, and solve them systematically, you can describe many real-life situations with clarity and confidence.
Keywords That Signal Linear Inequalities
Before solving any inequality, the first and most important step is to understand the language used in the question. Many questions do not directly tell you which inequality symbol to use; instead, they present information using English words that imply limits, ranges, or comparisons.
Learning to identify these keywords will help you instantly decide whether to use >, <, ≥, or ≤.
This skill is extremely important in examinations, especially for word problems.
Below is a detailed explanation of the most common keywords, what they mean, and how they must be interpreted.
Strictly Greater Than ( > )
These keywords mean the value must be more than the given number.
Equality is NOT allowed.
Keywords in this group:
▪ More than ▪ Greater than. ▪ Above
Examples:
The number of students is more than 30.
n > 30
The temperature must be above 25°C.
T > 25
Strictly Less Than ( < )
These keywords mean the value must be smaller than the given number.
Equality is NOT allowed.
Keywords in this group:
▪ Less than ▪ Lower than ▪ Under ▪ Below
Examples:
The time taken must be less than 10 minutes.
t < 10
The water level must stay below 3 cm.
h < 3
Greater Than or Equal To ( ≥ )
These keywords mean the value cannot fall below a certain limit.
Equality IS allowed.
Keywords in this group:
▪ At least ▪ Minimum ▪ No less than ▪ Greater than or equal to. ▪ Not below
Examples:
You need at least 75 marks to get a distinction.
s ≥ 75
The minimum height to ride the roller coaster is 120 cm.
h ≥ 120
Students should read for no less than 30 minutes daily.
t ≥ 30
Less Than or Equal To ( ≤ )
These keywords set an upper limit.
Equality IS allowed.
Many keywords fall under this group as they imply a boundary that must not be exceeded.
Keywords in this group:
▪ At most ▪ Maximum ▪ No more than ▪ Not exceeding ▪ Up to ▪ Less than or equal to
Examples:
You can bring at most 3 library books.
b ≤ 3
Your spending must not exceed 50 dollars.
x ≤ 50
The lift has a maximum load of 900 kg.
w ≤ 900
The temperature should stay up to 40°C.
T ≤ 40
“Between” Keywords
These refer to values within a range.
The inequality depends on whether the boundary is included.
Keywords in this group:
▪ Between A and B ▪ From A to B ▪ At least A but no more than B
Meanings:
Between A and B (strict):
A < x < B
Between A and B inclusive:
A ≤ x ≤ B
Examples:
The temperature must stay between 18°C and 25°C inclusive.
18 ≤ T ≤ 25
The reaction works only between 50°C and 80°C (strict).
50 < T < 80
Your weekly allowance will be at least 80 dollars but less than 120 dollars.
80 ≤ x < 120
Teacher’s Tip – The “crocodile rule”
One of the reasons why students find inequality confusing is that they cannot determine which side of the inequality is larger or small. To overcome this problem, you can imagine the inequality symbol to be a crocodile’s mouth, and that the crocodile will always “choose to eat” the larger number
Example 1 [ x < 5 ]
Since the crocodile “chooses to eat x“, then x is greater than 5. We interpret x > 5 as x is larger than 5.
Example 2 [ y < 9 ]
Since the crocodile “chooses to eat 9”, then 9 is greater than y. We interpret y < 9 as y is less than 9.
Note: The crocodile rule can also be used to determine the larger side with inequalities for ‘≤’ and ‘≥’.
| Symbol | Meaning of Symbol | Common Keywords | *Examples (Interpretations) *For easy learning, we only deal with integers here. |
| > | Strictly greater than |
| x > 10 (x is greater/ more than 10. x can be 11, 12, 13, 14 etc) |
| ≥ | Greater than or equal to |
| x ≥ –2 (x is greater than or equal to –2. The smallest value x can take on is –2. x can take on other values such as –2, –1, 0, 1, 2 etc) |
| < | Strictly less than |
| x < 3 (x is smaller/less than 3. x can take on values such as 2, 1, 0, –1, –2 etc) |
| ≤ | Less than or equal to |
| x ≤ 5 (x is less than or equal to 5. The largest value x can take on is 5. x can take on other values such as 4, 3, 2, 1, 0, –1 etc) |
< <
≤ ≤
< ≤
≤ < | Between |
| –5 < x < 5 (x is in between –5 to 5, but x does not include –5 nor 5. x can take on values such as –4, –3, –2, –1, 0 ,1 ,2 , 3 and 4 etc).
–5 ≤ x ≤ 5 (x is in between –5 to 5 inclusive. x can take on values such as –5, –4, –3, –2, –1, 0 ,1 ,2 , 3, 4 and 5 etc).
–5 < x ≤ 5 (x is in between –5 to 5, excluding –5 and including 5. x can take on values such as –4, –3, –2, –1, 0 ,1 ,2 , 3, 4 and 5 etc). –5 ≤ x < 5 (x is in between –5 to 5 including –5 and excluding 5. x can take on values such as –5, –4, –3, –2, –1, 0 ,1 ,2 , 3, and 4 etc). |
Rules for Solving Inequalities
When solving linear inequalities, almost all the rules you learn from linear equations still apply. You will still:
- Expand brackets the same way.
- Group like terms the same way, including factorisations.
- Balance the inequality by adding, subtracting, multiplying or dividing the same numbers on both sides.
- Simplify step by step just like equations.
Example 1
Solve 3(x + 2) − 5 < 16.
Solution:
Expand: 3x + 6 − 5 < 16
Simplify LHS: 3x + 1 < 16
Subtract 1 from both sides: 3x < 15
Divide both sides by 3: x < 5 (ans)
Example 2
Solve 5(2 – y) + 1 ≥ 3y – 5.
Solution:
Expand: 10 – 5y + 1 ≥ 3y – 5
Simplify LHS: 11 – 5y ≥ 3y – 5
Add 6 to both sides: 16 –5y ≥ 3y
Add 5y to both sides: 16 ≥ 8y
Divide both sides by 8: 2 ≥ y
*Reverse inequality: y ≤ 2 (ans)
*Generally we put the variable on the LHS. This is like writing y = 2 instead of 2 = y
As you can see from both examples shown above, inequalities behave exactly like equations in every situation except when carrying out a small handful of operations. The direction of the inequality sign “flips” under these special operations. We will go through two of such special cases.
Special Case 1 – Dividing by a negative number
The inequality sign “flips” when we divide both sides of the inequality by a negative number. This is the most common special case.
To illustrate the reasoning, we will use actual numbers to demonstrate.
Given: 1 < 10
When we divide both sides of the inequality by –1:
- value on LHS is –1
- value on RHS is – 10.
Note that –1 is larger than –10.
Hence we need to “flip” the inequality sign after dividing both sides by –1.
Given: 1 < 10
Divide both sides of the inequality by –1
and then flip inequality: –1 > – 10 [ –1 < – 10 (✗) ]
Caution: this is not the same as reversing inequality, as shown in Example 2 earlier (writing 2 ≥ y as y ≤ 2). Example Given: –3x – 6 > 15 Add 6 to both sides: –3x > 21 Divide both sides of the inequality by –1 and then flip inequality: x < 7 (ans)
Special Case 2 – Multiplying by a negative number
The inequality sign “flips” when we multiply both sides of the inequality by a negative number. This is because the larger number becomes “more negative” than the smaller number after the multiplication.
To illustrate this, we will use actual numbers to demonstrate.
Given: 5 ≥ 2
When we multiply both sides of the inequality by –3:
- value on LHS is –15
- value on RHS is – 6.
Note that –6 is larger than –15.
Hence we need to “flip” the inequality sign after multiplying both sides by –3.
Given: 5 ≥ 2
Multiply both sides of the inequality by –3
and then flip inequality: –15 ≤ – 6 [ –15 ≥ – 6 (✗) ]
Note: this is not the same as reversing inequality, as shown in Example 2 earlier.
Example
Given: − 1/2 x + 4 ≤ 10
Subtract 4 from both sides: − 1/2 x ≤ 6
Multiply both sides of the inequality by –2
and then flip inequality: x ≥ –12 (ans)
Further Example
Solve –3(x + 5) > 5(x + 5).
Solution:
Expand: –3x – 15 > 5x + 25
(There is no flipping here since we did not multiply or divide the same negative number on both sides.)
Add 15 to both sides: –3x > 5x + 40
Subtract 5x from both sides: –8x > 40
(There is no flipping here since we did not multiply or divide the same negative e number on both sides.)
Divide both sides by –8 and flip inequality sign: x < –5 (ans)
Representing Solution on Number Line
The solution set to a linear inequality can be represented on a number line with the symbols below.
open dot for < and > ( means the endpoint value is not included )
solid dot for ≤ and ≥ ( means the endpoint value is included )
arrows
Inequality notation Number line
Examples
(1) The solution set for x ≥ 3 is as follow.
This means all real numbers greater than or equal to 3.
(2) The solution set for x < ‒ 5 is as follow.
This means all real numbers less than ‒5.
(3) The solution set for –2 ≤ x < 4 is as follow.
This means all real numbers from –2 inclusive to real numbers less than 4.
Solving Simultaneous Linear Inequalities
Simultaneous inequalities involve two or more conditions that x must satisfy at the same time.
The final answer is where the ranges overlap.
Example 1
Solve the following inequalities.
2x − 3 > 5 and x + 4 ≤ 12
Solution:
Solve each inequality separately:
2x − 3 > 5 x + 4 ≤ 12
2x > 8 x ≤ 8
x > 4
Combine both:
x > 4 and x ≤ 8 ∴ 4 < x ≤ 8 (ans)
Some students have difficulty in combining inequalities. An effective way to overcome this problem is to use the number line to represent both solution sets on the same number line and look for overlapping areas. If there are no overlapping areas, then the simultaneous inequalities have no solution.
See that the overlapping area is in between 4 and 8. Do remember to use the correct inequality sign when writing the final answer.
Example 2
Solve the following simultaneous inequalities.
−10 < 5 − 3x ≤ 2
Solution:
Solve each inequality separately:
−10 < 5 − 3x and 5 − 3x ≤ 2
−15 < −3x −3x ≤ −3
5 > x x ≥ 1
x < 5
Combining the inequalities, we have 1 ≤ x < 5 (ans).
Common Mistakes When Working With Inequalities
1. Forgetting to flip the inequality when dividing by a negative value.
This is the number one cause of errors.
Eg. –3x > 30
x < –10 (✓) x > –10 (✗)
2. Confusing < with ≤ or > with ≥
Students must check if equality is allowed or specified.
Eg. State the largest prime number given x < 11.
Ans: 11 (✗) Ans: 7 (✓)
3. Drawing wrong circles on the number line
Open circle when endpoint is not included.
Closed circle when endpoint is included.
Eg. Given x ≥ –3 and x ≤ 5. Represent the solution set on a number line.
4. Giving a single answer instead of a range
Eg. The cost of a mobile phone, c, does not exceed $500. Represent the statement as an inequality in terms of c.
Ans: c = 499 (✗) Ans: c < 500 (✓)
5. “Flipping” the inequality direction when adding or subtracting
Adding and subtracting never “flips” the sign.
Eg. 5x + 4 > 14
5x < 10 (✗) 5x > 10 (✓)
6. Indicating the wrong direction on number line
Students must always check whether the solution increases to the right or decreases to the left.
Eg. Given 2 > x. Represent this solution set on a number line.
Note that “2 > x” reverses to “x < 2″.
7. Reversing terms incorrectly.
Eg. 12 – 2x > x
12 > 3x
4 > x
x > 4 (✗) x < 4 (✓)
Conclusion
Linear inequalities are an essential part of algebra because they help us describe limits, boundaries, and ranges of possible values. In many real-life situations, we are not looking for one exact answer but a set of values that meet certain conditions.
By learning to recognise keywords, interpret inequality symbols, apply the correct solving rules, and represent solutions on a number line, you can approach inequality questions with confidence.
With practice, you will find that inequalities are simply an extension of the equation skills you already know, giving you a powerful tool to analyse and model real-world situations.
Frequently Asked Questions
What is a linear inequality?
A linear inequality is similar to a linear equation, but instead of an equal sign, it uses a comparison symbol such as >, <, ≥ or ≤. While an equation like 2x + 3 = 7 has one exact solution, a linear inequality like 2x + 3 > 7 has a whole range of solutions.
How do I know when to flip the inequality sign?
You only flip the inequality sign when you multiply or divide both sides of the inequality by a negative number. Adding or subtracting a number, even a negative one, never flips the sign. This is the single most common mistake students make with linear inequalities.
What is the difference between an equation and an inequality?
An equation has exactly one solution, while an inequality has a range of possible solutions. For example, x = 5 has only one answer, but x > 5 is true for infinitely many values, such as 6, 7, 10 or 100.
How do you solve simultaneous linear inequalities?
Solve each inequality separately, then find where the two solution sets overlap. Using a number line to plot both solution sets side by side makes it easier to see the overlapping range, which becomes the final answer.
How do you show a solution set on a number line?
Use an open circle for < or > to show the endpoint is not included, and a solid circle for ≤ or ≥ to show the endpoint is included. Then draw an arrow in the direction of all the values that satisfy the inequality.
Before you go, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
Does your child need help with Mathematics?
Find out more about our Math Tuition Class Free Trial.
1) Live Zoom Lessons at Grade Solution Learning Centre
At Grade Solution Learning Centre, we are a team of dedicated educators whose mission is to guide your child to academic success. Here are the services we provide:
– Live Zoom lessons
– Adaptably, a smart learning platform that tracks your child’s progress, strengths and weaknesses through personalised digital questions.
– 24/7 Homework Helper Service
We provide all these services above at a very affordable monthly fee to allow as many students as possible to access such learning opportunities. We also offer a free trial class and consultation with our tutors, to help make sure we are the best fit for your child.
We specialise in English, Math, Science, and Chinese subjects.
You can see our fees and schedules here >>
2) Pre-recorded Online courses on Jimmymaths.com
If you are looking for something that fits your budget, or prefer your child learn at his or her own pace, you can join our pre-recorded online Math courses.
Your child can:
– Learn from recorded videos
– Get access to lots of common exam questions to ensure sufficient practice
– Get unlimited support and homework help
You can see the available courses here >>