Inequalities.

Step 1 of 4

Where is it smaller?

An inequality does not ask when the two sides are equal, but when one is smaller or larger than the other: $2x + 1 < 7$⁠. On the board the question is where the blue line, the left side, lies below the pink line, the right side.

The meeting point, $x = 3$⁠, is the boundary. On one side of it the blue line is below the pink one, and on the other side above it. Predict which side the solution is on, and then the sign < will colour the solution on the $x$⁠-axis.

Step 2 of 4

Multiplying by a negative number

You solve $2x + 1 < 7$⁠ just like an equation: subtract $1$⁠ from both sides and divide by $2$⁠, which gives $x < 3$⁠. But there is one new rule: when you multiply or divide both sides by a negative number, the sign turns round.

Why? $2 < 5$⁠, but $-2 > -5$⁠: multiplying by $-1$⁠ moves every number to the other side of zero, and the order reverses. The board now shows $-2x + 1 < 7$⁠. The blue line slopes down, so it is below the pink line on the right-hand side instead.

Step 3 of 4

The boundary itself

In $x < 3$⁠ the boundary $3$⁠ itself is not a solution: $2 \cdot 3 + 1 = 7$⁠, and $7$⁠ is not less than $7$⁠. On the board it is marked with an empty ring. In $x \le 3$⁠ (less than or equal to) it is a solution, marked with a filled dot. Switch Sign between < and ≤ and see.

Step 4 of 4

Your turn

The board shows $5 - x > 2$⁠, so the left side is $-x + 5$⁠. Solve it yourself, then pick > under Sign to check on the $x$⁠-axis.