   SEARCH HOME Math Central Quandaries & Queries  Question from Hempstone, a student: Solve the inequalities 3x-5<2x+5>4x+9 Hi,

First of all you shouldn't write a string of inequalities on the same line with the inequalities pointing in different directions. At first glance it seems like there is a relationship between 3x-5 and 4x+9, but there isn't. You should write it as two inequalities

$3x - 5 < 2x + 5 \mbox{ and } 2x + 5 > 4x + 9.$

To solve the problem look at the two inequalities separately. I am going to work with your second equation and write it as

$4x+9 < 2x + 5.$

You can manipulate inequalities as you would equations with some provisos. You can add or subtract the same number to each side and the direction of the inequality remains unchanged. So if you add -9 to both sides of the inequality you get

$4x < 2x - 4.$

Now adding -2x to both sides gives

$2x < -4.$

I want $x$ alone on the left side so I want to divide both sides by 2 or, equivalently, multiply both sides by $\large \frac{1}{2}.$ t is at this point you need to take special care if you are working with inequalities. If you multiply or divide both sides on an inequality by a positive number then the direction of the inequality remains unchanged. If you multiply or divide both sides on an inequality by a negative number then the direction of the inequality reverses. Hence multiplying both sides by $\large \frac{1}{2}$ yields

$x < -2.$

Thus the solution to your second inequality is all numbers less than -2.

Try this technique on your first equation and the solution to the pair of inequalities is all numbers that satisfy both inequalities.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.