Posted in inequalities, puzzles.

The square root of 2 is 1.41421356237... Multiply this successively by 1, by 2, by 3, and so on, writing down each result without its fractional part: 1 2 4 5 7 8 9 11 12... Beneath this, make a list of the numbers that are missing from the first

Read More →
Posted in ask uncle colin, inequalities.

Dear Uncle Colin, I'm pretty good with quadratic inequalities and pretty good with absolute values, but when I get the two together, I get confused. For example, I struggled with the set of values satisfying $x^2 -\left| 5x-3\right| < 2 + x$. Can you help? - Nasty Absolute Value Inequalities

Read More →
Posted in ask uncle colin, inequalities.

Dear Uncle Colin, I've come across a seemingly simple question I can't tackle: solve $x^2 + 2x \ge 2$. I tried factorising to get $x(x+2) \ge 2$, which has the roots 0 and -2, but the book says the answer is $x < -1-\sqrt{3}$ or $x > -1 + \sqrt{3}$.

Read More →
Posted in core 3, inequalities, modulus.

Solve $\sqrt{|x|-3} > x-4$ Difficult, as the man said. Difficult, difficult, lemon difficult. It’s not that it’s tricky to solve it - it’s just… fiddly. Let’s start by drawing the graphs. The right-hand side is easy enough: it’s a straight line with gradient 1, through the point $(0,-4)$. The left-hand

Read More →