Free maths tool · Level: Ages 13 to 15
Solving inequalities
Slide the numbers and watch which stretch of the number line lights up. Set the number in front of the x negative and see what the sign does.
An inequality is solved just like an equation, with one difference: multiply or divide both sides by a negative number and the sign flips. For -2x < 6 you divide by -2 and it becomes x > -3. Adding and subtracting never change the sign. The answer is not a single number but a whole stretch of the number line.
The answer is a stretch of the line
An equation looks for one number. An inequality looks for every number that works, and usually there are infinitely many. So the answer is drawn as a shaded stretch of the number line rather than as a single point.
Watch the dot at the boundary. With < or > the boundary itself is not a solution and the dot stays hollow. With ≤ or ≥ it belongs to the answer and the dot is filled in. That small difference really is marked.
Why the sign flips
Take 2 < 5, which is true. Multiply both sides by -1 and you get -2 and -5. Now -2 is the bigger one, because on the number line -2 sits to the right of -5. Multiplying by a negative reflects everything about zero, and a reflection reverses every order.
That is why the sign flips exactly when you divide or multiply by a negative, and at no other moment. Set the number in front of the x negative and the shaded side does indeed swap over.
Solving it graphically
Switch the graph on and two lines appear: the sloping y = ax + b and the flat y = c. The inequality asks where the sloping one is below or above the flat one, and that is exactly the stretch shaded on the line underneath.
The two lines cross at the boundary, and either side of it one of them is on top. That shows the boundary comes from the equation, and the inequality only says which side you want.
Is that flipping sign still confusing?
A tutor practises it with you until you can see straight away when it applies.
Frequently asked questions
- When does an inequality sign flip?
- Only when you multiply or divide both sides by a negative number. Adding, subtracting and dividing by a positive all leave it alone. That is the single rule that differs from an ordinary equation.
- How do you solve -2x < 6?
- Divide both sides by -2. Since -2 is negative the sign flips and you get x > -3. The answer is everything to the right of -3, with a hollow dot because -3 itself does not count.
- What is the difference between a hollow and a filled dot?
- A hollow dot means the boundary is excluded, which goes with < and >. A filled dot means it is included, which goes with ≤ and ≥.
- How do you solve an inequality graphically?
- Draw both sides as functions and see where one graph is above or below the other. The crossing point gives the boundary, and the sign you were given says which side of it is the answer.
- How do you write the solution set?
- Usually as an interval, for example ]-3, +∞[ for x > -3. A square bracket means the end value is included, a round one that it is not.
More interactive maths tools
- Solving equations →Solve with the balance method and see why the same move goes on both sides.
- Slope of a line →Drag two points and see Δy/Δx as a staircase, with the equation and the perpendicular alongside.
- System of equations →Two lines, one crossing point, with the opening steps of all three solving methods.