Solving the Inequality ( frac{x-1}{x-2} geq 4 )

In this article, we will analyze and solve the inequality ( frac{x-1}{x-2} geq 4 ). To solve this problem, we will identify the critical points and divide the number line into intervals based on these points. This method helps us understand the behavior of the inequality in different regions and ultimately find the solution set.

Analysis of the Inequality

The inequality we are dealing with is ( frac{x-1}{x-2} geq 4 ). To analyze this, we consider the critical points where the expressions within the fraction change sign. These points are ( x 1 ) and ( x 2 ).

Intervals Divided by Critical Points

These critical points divide the number line into three intervals:

( x ( 1 leq x ( x geq 2 )

Interval 1: ( x

In this interval, both ( x - 1 ) and ( x - 2 ) are negative. We express them as:

( x - 1 - (x - 1) 1 - x ) ( x - 2 - (x - 2) 2 - x )

Substituting these into the inequality, we get:

[ 1 - x geq 2 - x quad Rightarrow quad 1 geq 4 ]

This is a contradiction, meaning there are no solutions in this interval.

Interval 2: ( 1 leq x

In this interval, ( x - 1 ) is non-negative, and ( x - 2 ) is negative. We express them as:

( x - 1 x - 1 ) ( x - 2 2 - x )

Substituting these into the inequality, we get:

[ x - 1 geq 2 - x quad Rightarrow quad 2x geq 3 quad Rightarrow quad x geq frac{3}{2} ]

Since ( 1 leq x

Interval 3: ( x geq 2 )

In this interval, both ( x - 1 ) and ( x - 2 ) are non-negative. We express them as:

( x - 1 x - 1 ) ( x - 2 x - 2 )

Substituting these into the inequality, we get:

[ x - 1 leq x - 2 quad Rightarrow quad -1 leq -2 ]

This is always false, meaning there are no solutions in this interval.

Summary of Solutions

Combining the results from all intervals, we find:

No solutions in ( x Solutions in ( frac{3}{2} leq x No solutions in ( x geq 2 )

Therefore, the solution to the inequality ( frac{x-1}{x-2} geq 4 ) is ( x geq frac{3}{2} ).

Conclusion

By analyzing the inequality through its critical points and intervals, we were able to determine that the solution set is ( x geq frac{3}{2} ). This method is applicable for similar inequalities involving fractions and absolute values.