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2) |2x-3|-|5-x|-|x-10|-|3-x|=28

3) -|2x-3|+|5-x|+|x-10|≥|3-x|

How can we solve these problems?

The method I know is to plug in the critical values to see which modulus becomes positive and which one becomes negative. Then find out the values of x for which the above conditions become true.

For example, if I am asked to solve the first problem, I will do the following steps.

Step-1: Find out the critical values.

For 2x-3, critical value is 3/2.

For 5-x, critical value is 5.

For x-10, critical value is 10

For 3-x, critical value is 3.

Step-2: When x is less than 3/2, we can write the first equation in the following form:

(2x-3)-(x-5)-(x-10)=-(x-3)

or, 2x-3-x+5-x+10=-x+3

or, x=-9

When 1.5≤x<3,

-(2x-3)-(x-5)-(x-10)=-(x-3)

or, -2x+3-x+5-x+10=-x+3

or, -4x+18=-x+3

or, -3x=-15

or, x=5

x=5 isn't acceptable because we assumed x<3.

We can repeat the step-2 for 3≤x<5, 5≤x<10 and x≥10.

This procedure is very long and a time consumable process.

Is there other way that requires less time and less effort to solve the above problems?

Once I have seen a person to use a number line to solve this sort of problem. However, his explanation works when there is a poisitive sign between two modulus functions, and a value at the right hand side of the equation.

For example, we can consider the following equation.

4) |x-5|+|x-2|=9

We can imagine a number line where two people are standing at two points. One person is 2 units away from zero (center point), and other person is 5 units away from zero. Both persons are standing at the right hand side of center point. They want to meet on such a point for which together they will have to travel total 9 units.

So if the person who is two units away from center point moves right, they can meet on a point which is 8 units away from center point. Hence total distance traveled by them will be now 9 units. So, now we have x=8.

If we repeat this same procedure for the left hand side, we will find x=-1 which is just one unit away from zero (center point). And to reach that point, both of them will have to travel total 9 units.

However, this procedure doesn't work for the equations like 1,2 and 3.

Now I am here to ask the experts of this site to let me know some other ways which can help solve the above equations within shortest possible time.