Inequalities Worked Solutions — A-Level Maths

Subtract 4 from both sides

We want the term with x on its own, so we undo the +4 by subtracting 4 from each side. Because we are only adding or subtracting, the inequality sign stays exactly the same.

Divide both sides by 3

Now we undo the multiplication by 3 by dividing both sides by 3. We are dividing by a positive number, so the inequality sign does not flip.

State the solution

So every number bigger than 5 makes the inequality true. On the number line we use an open circle at 5 to show 5 itself is not included.

Add 7 to both sides

To get the x-term by itself we undo the -7 by adding 7 to both sides. Adding the same amount to each side never changes the direction of the inequality.

Divide both sides by 5

We divide by 5 to leave x alone. Since 5 is positive, the sign stays as 'less than or equal to'.

State the solution

All values up to and including 5 work. We show this with a filled (solid) circle at 5 because 5 is included.

Identify what to divide by

The x is multiplied by -2, so to free it we must divide both sides by -2. This is the key step where care is needed.

Divide by -2 and flip the sign

Whenever you multiply or divide an inequality by a NEGATIVE number, you must reverse the inequality sign. So '>' becomes '<'.

State the solution

The answer is all numbers less than -4. Forgetting to flip the sign here is the most common mistake, so always check the direction.

Subtract 2x from both sides

There are x-terms on both sides, so gather them on one side. Taking 2x from each side keeps the inequality balanced and the sign unchanged.

Subtract 1 from both sides

Now move the number term away from the x-term by subtracting 1 from both sides.

Divide both sides by 2

Dividing by the positive number 2 finishes the job and the sign stays the same.

Expand the bracket

Multiply everything inside the bracket by 3: 3 times x is 3x and 3 times -2 is -6. This clears the bracket so we can solve normally.

Add 6 to both sides

Undo the -6 by adding 6 to both sides to isolate the x-term.

Divide both sides by 3

Dividing by the positive number 3 gives the final answer, with the sign kept as 'greater than or equal to'.