I’ll try to help you with an easy-to-understand and more visual approach to solve this question.
Quickly plot the 2 lines on x-y axes:
Y= 3x-2
Y= 2x-5
We find that they intersect in the 3rd Quadrant.
Attachment:
File comment: Graphical Method of solving Mod Inequalities
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Now comes
the key point that you must understand: what does |3x-2| mean?
For any given value of x, the modulus sign |3x-2| denotes the distance of the point (3x-2) from the x-axis.Once we understand this, the problem becomes pretty easy to solve.
We can observe from the graph that for x<-3, the line 3x-2 is at a greater distance from the x-axis than the line 2x-5. Or, to put it mathematically,
|3x-2|>|2x-5|
At x=-3, both lines are at the same distance from the x-axis. i.e. |3x-2|= |2x-5|
As we move rightwards from x=-3, we observe that the line 3x-2 is closer to the x-axis than the line 2x-5. How do we say it mathematically? Yeah, you guessed it:
|3x-2|<|2x-5|
Tada! This is the zone we wanted to be in!
But, is this true for all values of x>-3? Again, take a look at the graph.
We observe that, after some point, let’s call it Point G, in the 1st Quadrant, the line 3x-2 again starts getting more distant from the x-axis than the line 2x-5.
How do we find this point G?
At the point G, both the lines will be equidistant from the x-axis; one will be above the x-axis, and the other will be below it. To put it in other words, the magnitude of y (y= 3x-2 and y= 2x-5) will be same for both lines at the point G; but the signs of y will be opposite.
So, the following equation will be true for point G:
3x-2 = -(2x-5)
Solving it, we get x= 7/5.
So, we can conclude that between x>-3 and x<7/5, the line 3x-2 is closer to the x-axis than the line 2x-5. We say it mathematically as under:
|3x-2|<= |2x-5| for -3<=x<=7/5