### Weekly Challenge 43: A Close Match

Can you massage the parameters of these curves to make them match as closely as possible?

### Weekly Challenge 44: Prime Counter

A weekly challenge concerning prime numbers.

### The Right Volume

Can you rotate a curve to make a volume of 1?

# Inner Equality

##### Stage: 5 Short Challenge Level:

Inequalities are an important extension of algebra which are needed more formally in C1 and beyond.

Note that some, but not all, algebraic manipulations still work with equals signs 'replaced' by inequality signs -- you need to take extra care when algebraically manipulating inequalities.

Addition or subtraction of a quantity is straightforward with inequalities. For example, if we know that $5< a+b< 10$ then we know that $5-b< a< 10-b$. However, we need to take more care with division and multiplication as minus signs cause inequalities to reverse under these operations.

Direct algebra will not help you much in this problem. You will have to make deductions such as 'if a is a very small positive number than b must be very close to 5'.

Writing down such statements is difficult to do clearly, so focus on the inequalities intuitively if need be.