Hi Rosie,
I want to multiply both sides of the inequality by $x2$ but I know that if you multiply both sides of an inequality by a negative number the inequality changes direction. Thus I am going to have to separate the problem into two cases, $x  2 > 0$ and $x  2 < 0.$
 Case 1: $x  2 > 0$ which is $x > 2.$
 Since $x  2 > 0$ if I multiply both sides of $\frac{4x}{x  2} \leq 3$ by $x  2$ I get $4x \leq 3(x  2)$ which simplifies to $x \leq 6.$ Hence for case 1 the inequality is true if $x > 2$ AND $x \leq 6.$ But $x$ can't be both larger than 2 and less than or equal to 6 and hence this case found no $x$ that satisfies the inequality.

 Case 2: $x  2 < 0$ which is $x < 2.$
 Since $x  2 < 0$ if I multiply both sides of the inequality by $x  2$ the inequality becomes $\frac{4x}{x  2} \geq 3.$ Now you complete the problem.
Penny
