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 Question from Rosie, a student: 4x/x-2<3 four x divided by x-2 is less than or equal to 3

Hi Rosie,

I want to multiply both sides of the inequality by $x-2$ 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

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