   SEARCH HOME Math Central Quandaries & Queries  Question from Helen, a student: solve the following inequality for x -1<5-2x<10 The less then sign by the one, should be less then or equal to. We have two responses for you

Hi Helen,

Let me illustrate with a similar problem.

solve the following inequality for x
-6 < 4 - 5x ≤ 9

You can solve an inequality as you would solve an equality (equation) with one change. If you multiply or divide both sides of the inequality by a negative number, the direction of the inequality reverses. In my problem, as in yours, there are two inequalities, -6 < 4 - 5x and 4 - 5x ≤ 9 but if you are careful you can solve both at the same time.

The object is to manipulate the inequalities and end with a statement of the form

first number < x < second number

-6 < 4 - 5x ≤ 9

First I add -4 to both sides of each inequality to get

-6 -4 < 4 - 5x -4 ≤ 9 -4

or

-10 < - 5x ≤ 5

I don't want the negative sign between the inequalities so I multiply through by -1. Since -1 is negative this reverses the direction of the inequalities.

10 > 5x ≥ -5.

Since the number line is arranged so that larger numbers are to the right you might find it more natural to write this

-5 < 5x ≤ 10.

The final step is to divide through by 5 to get

-1 < x ≤ 2.

Penny

Helen, to answer these questions, you simply treat it like an equation
with three sides (whatever you do to one side, you have to do to the
other two as well) except that multiplying or dividing by a negative
causes the direction of the inequalities to change.

Here's an example:

20 > 7-4x > 4

I want x by itself in the middle, so first I subtract 7 from all sides:
20 - 7 > 7 - 4x - 7 > 4 - 7
13 > -4x > -3

Now I divide all sides by -4. I have to flip the direction of the
inequality because I am dividing by a negative:
13 / (-4) < -4x / (-4) < -3 / (-4)
-13/4 < x < 3/4

So x is between -13/4 and 3/4. I can write this as (-13/4, 3/4)

Now you try it with -1 < 5-2x < 10.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.