Solving Linear Inequalities
From speed limits to age restrictions, inequalities are all around us. In maths, they help us define a range of possible values. This guide will show you how to solve linear inequalities and present your answers on a number line.
The Rules of Solving Inequalities
Solving inequalities is just like solving equations, with one special rule to remember. Your goal is to get the variable (e.g., $x$) on its own.
Rules 1 & 2: The Basics
You can add or subtract any number from both sides, and the inequality symbol stays the same.
You can multiply or divide both sides by any positive number, and the symbol stays the same.
Rule 3: The Golden Rule!
If you multiply or divide both sides by a negative number, you MUST FLIP the inequality symbol.
Example: We know $6 > 4$.
Divide by -2: $\frac{6}{-2}$ and $\frac{4}{-2}$ becomes $-3$ and $-2$.
Now, $-3$ is less than $-2$, so we must flip the sign: $-3 < -2$.
Representing Solutions on a Number Line
A number line is a clear way to show the range of possible solutions.
Open Circle ($<$ or $>$)
Used for strict inequalities. It means the number itself is not included in the solution.
$x > 2$
Closed Circle ($\le$ or $\ge$)
Used for non-strict inequalities. It means the number itself is included in the solution.
$x \le -1$
Worked Examples
Example 1: Dividing by a Negative
Solve $-2x \ge 8$.
- To isolate $x$, divide both sides by -2.
- Because we are dividing by a negative number, we must flip the inequality sign from $\ge$ to $\le$.
- $\frac{-2x}{-2} \le \frac{8}{-2}$.
Answer: $x \le -4$
Example 2: Variables on Both Sides
Solve $5x + 2 > 3x + 10$.
- Collect $x$ terms on one side. Subtract $3x$ from both sides: $2x + 2 > 10$.
- Move constant terms. Subtract 2 from both sides: $2x > 8$.
- Isolate $x$. Divide by 2 (a positive number, so no sign flip).
Answer: $x > 4$
Tutor Insights
🤔 Common Misunderstandings
- Forgetting to flip the sign. This is the most common mistake by far. Always be alert when multiplying or dividing by a negative.
- Confusing open and closed circles. A good tip: if the symbol has a line underneath ($\le, \ge$), the circle is filled in.
📝 Common Exam Mistakes
- Sign errors during rearrangement.
- Incorrect circle type or arrow direction on the number line.
- Not labelling the number line with the correct boundary value.
Practice Questions
Solve each inequality and represent the solution on a number line.
- $3x \ge 21$
- $-4x > 8$
- $10 – x < 3$
- $3(x – 2) \le x + 4$
Show Answers
- $x \ge 7$
- $x < -2$
- $x > 7$
- $x \le 5$
FAQs
What’s the main difference between solving equations and inequalities?
The main difference is the special rule for inequalities: if you multiply or divide both sides by a negative number, you must flip the inequality symbol ($<$ becomes $>$, $\ge$ becomes $\le$, etc.). Also, equations usually have one specific solution, while inequalities have a range of solutions.
How do I remember if I need an open or closed circle?
Think of the line under the inequality symbol. If there’s a line ($\le$ or $\ge$), it means “or equal to”, so the number is included and the circle is closed (filled in). If there’s no line ($<$ or $>$), the number is not included, so the circle is open.
