Linear Inequalities (GCSE Maths)
Skill Check
Solve the following inequality:
3(n + 1) < 24
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Divide both sides by 3
This isolates the expression inside the brackets:
Step 2 · Subtract 1 from both sides
This isolates the variable $n$:
Step 3 · Consider the alternative method
You can also solve this by expanding the brackets first:
- Expand the brackets to get $3n + 3 < 24$.
- Subtract $3$ to get $3n < 21$.
- Finally, divide by $3$ to get $n < 7$.
Final Answer
$n < 7$
Solve the following inequality:
2(3n − 5) > 12
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Expand the brackets
Multiply both terms inside the brackets by $2$:
Step 2 · Add $10$ to both sides
This isolates the term containing $n$:
Step 3 · Divide both sides by $6$
Step 4 · Simplify the fraction
Divide the numerator and denominator by $2$ (or convert to a mixed number):
Final Answer
$n > \frac{11}{3}$
Write down the inequality shown on the number line.
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Identify the variable
Notice that the number line is labelled with the variable $y$ on the far right.
Step 2 · Examine the lower bound
- There is a solid (filled) circle at $-3$.
- A solid circle indicates that the number is included in the range, meaning we use the "less than or equal to" symbol ($\le$).
Step 3 · Examine the upper bound
- There is also a solid (filled) circle at $4$.
- This indicates that $4$ is included in the range as well, meaning we again use the "less than or equal to" symbol ($\le$).
Step 4 · Combine into a compound inequality
The line connects $-3$ and $4$, meaning $y$ represents all values between and including $-3$ and $4$. Therefore, the inequality is:
Final Answer
$-3 \le y \le 4$
Write down the inequality shown on the number line.
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Identify the variable
Notice that the number line is labelled with the variable $y$ on the right side.
Step 2 · Examine the lower bound
- There is a solid (filled) circle at $-4$.
- This indicates that $-4$ is included in the range, meaning we use the "less than or equal to" symbol ($\le$).
Step 3 · Examine the upper bound
- There is an open (unfilled) circle at $3$.
- This indicates that $3$ is not included in the range, meaning we use the strictly "less than" symbol ($<$).
Step 4 · Combine into a compound inequality
The line connects $-4$ and $3$, meaning $y$ takes all values between them (including $-4$, but excluding $3$). Therefore, the inequality is:
Final Answer
$-4 \le y < 3$
Write down the inequality shown on the number line.
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Identify the circle's position
The circle is positioned above the number $2$ on the number line.
Step 2 · Determine the type of inequality
- The circle is open (unfilled), which means the inequality does not include the number $2$ itself.
- Therefore, we use strictly less than ($<$) or strictly greater than ($>$).
Step 3 · Determine the direction of the arrow
The arrow points to the left towards smaller numbers (negative infinity).
Step 4 · Write the inequality
Since the arrow points to all numbers strictly less than $2$, the inequality is:
Final Answer
$x < 2$
Problem Solving
Solve the inequality: 2x + 9 > 19 − 8x
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Add $8x$ to both sides
This brings all the terms containing $x$ to the left-hand side:
Step 2 · Subtract $9$ from both sides
This isolates the $10x$ term on the left-hand side:
Step 3 · Divide both sides by $10$
This gives the final solution for $x$:
Final Answer
$x > 1$
Solve the inequality below and represent the solution on a number line: $$\frac{x + 18}{4} \le 5$$
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Multiply both sides by 4
This removes the fraction:
Step 2 · Subtract 18 from both sides
Step 3 · Represent on a number line
- Draw a solid (filled) circle at $2$ on the number line to show that $2$ is included.
- Draw an arrow extending from the circle to the left (towards smaller numbers / negative infinity).
Final Answer
$x \le 2$
Solve the inequality below and represent the solution on a number line: $$\frac{x}{2} - 3 > 0$$
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Add $3$ to both sides
Step 2 · Multiply both sides by $2$
Step 3 · Represent on a number line
- Draw an open (unfilled) circle at $6$ on the number line (to show that $6$ is not included).
- Draw an arrow extending from the circle to the right (towards larger numbers/positive infinity).
Final Answer
$x > 6$
Solve the inequality below and represent the solution on a number line: 3x - 2 ≥ 10
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Add $2$ to both sides
Step 2 · Divide both sides by $3$
Step 3 · Represent on a number line
- Draw a solid (filled) circle at $4$ on the number line (to show that $4$ is included).
- Draw an arrow extending from the circle to the right (towards positive infinity).
Final Answer
$x \ge 4$
Solve the inequality below and represent the solution on a number line: 4x + 7 < 11
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Subtract 7 from both sides
Step 2 · Divide both sides by 4
Step 3 · Represent on a number line
- Draw an open (unfilled) circle at $1$ on the number line to show that $1$ is not included.
- Draw an arrow extending from the circle to the left towards smaller numbers.
Final Answer
$x < 1$
Exam-Style Questions
Solve each of the inequalities below:
13x − 12 < 3x + 13
7x − 5 ≥ 3x + 11
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Subtract $3x$ from both sides
Group the variable terms on the left:
Step 2 · Add 12 to both sides
Isolate the $10x$ term:
Step 3 · Divide both sides by 10
Final Answer
$x < 2.5 \quad \text{or} \quad x < \frac{5}{2}$
Step 1 · Subtract $3x$ from both sides
Group the variable terms on the left:
Step 2 · Add 5 to both sides
Step 3 · Divide both sides by 4
Final Answer
$x \ge 4$
Solve each of the inequalities below:
4x + 3 > 2x + 11
x + 1 ≥ 3x − 18
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Subtract $2x$ from both sides
Bring the variable terms to one side:
Step 2 · Subtract $3$ from both sides
Isolate the term containing $x$:
Step 3 · Divide both sides by $2$
Final Answer
$x > 4$
Step 1 · Add $18$ to both sides
Bring the constant terms to the left-hand side:
Step 2 · Subtract $x$ from both sides
Keep the $x$ term positive on the right-hand side:
Step 3 · Divide both sides by $2$
Final Answer
$x \le 9.5$
On the number line, show the inequality x + 1 ≤ 4.
5 < 2y < 12 where y is an integer.
Write down all the possible values of y.
Solve 4 > 19 − 3x.
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Solve for $x$
Subtract $1$ from both sides of the inequality:
Step 2 · Identify the circle type
Because the inequality uses the "less than or equal to" symbol ($\le$), place a solid (filled) circle above $3$ on the number line.
Step 3 · Determine the direction
Since $x$ is less than or equal to $3$, draw an arrow extending from the circle to the left (towards smaller numbers/negative infinity).
Final Answer
$x \le 3$
Step 1 · Isolate the variable $y$
Divide the entire compound inequality by $2$:
Step 2 · Identify the boundaries
Since $y$ must be strictly greater than $2.5$ and strictly less than $6$, the integer $3$ is the smallest possible value and $5$ is the largest possible value ($6$ is not included).
Step 3 · List the integers
The integer values of $y$ within this range are:
Final Answer
$3, 4, 5$
Step 1 · Add $3x$ to both sides
Bring the variable term to the left-hand side so it becomes positive:
Step 2 · Subtract $4$ from both sides
Step 3 · Divide both sides by $3$
Final Answer
$x > 5$
On the number line, show the inequality n < 2.
4 ≤ y < 8 where y is an integer.
Write down all the possible values of y.
Solve 4x + 6 ≤ x + 21.
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Identify the circle type
The inequality uses the strictly less than symbol ($<$), so place an open (unfilled) circle above $2$ on the number line.
Step 2 · Determine the direction
Since $n$ is less than $2$, draw a line or arrow extending from the circle to the left (towards smaller numbers).
Final Answer
$\text{Open circle at } 2\text{, arrow to the left}$
Step 1 · Identify the bounds
- The lower bound symbol $\le$ means that $4$ is included in the list of values.
- The upper bound symbol $<$ means that $8$ is not included in the list of values.
Step 2 · List the integers
The whole numbers within this range are $4, 5, 6, 7$.
Final Answer
$4, 5, 6, 7$
Step 1 · Subtract $x$ from both sides
Step 2 · Subtract 6 from both sides
Step 3 · Divide both sides by 3
Final Answer
$x \le 5$
On the number line, show the inequality x > −3.
1 ≤ y < 5 where y is an integer.
Write down all the possible values of y.
Solve 4t + 7 ≤ 19.
- Take a clear photo of your handwritten work.
- Ensure all calculation steps are legible.
- Upload below for AI Tutor analysis.
Step 1 · Describe the number line for part
- The inequality uses the strictly greater than symbol ($>$), so an open (unfilled) circle is placed above $-3$.
- Since $x$ is greater than $-3$, draw an arrow extending from the circle to the right.
Step 2 · List the integers
- The lower bound symbol $\le$ means that $1$ is included in the list of values.
- The upper bound symbol $<$ means that $5$ is not included in the list of values.
- The whole numbers within this range are $1, 2, 3, 4$.
Step 3 · Solve the linear inequality
Subtract $7$ from both sides:
Divide both sides by $4$:
Final Answer
(a) $\text{Open circle at } -3\text{, arrow to the right}$ (b) $1, 2, 3, 4$ (c) $t \le 3$
Feel confident with Linear Inequalities (GCSE Maths)?
Review the master revision notes or move on to the next topic.