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To solve the inequality [tex]\( |x-3| < |x-2| \)[/tex], we can follow a step-by-step approach:
1. Understand the Inequality: The inequality involves absolute values, which means it looks at the distance of [tex]\( x \)[/tex] from 3 and [tex]\( x \)[/tex] from 2 and states that the distance from 3 is less than the distance from 2.
2. Consider Different Cases: Absolute value inequalities can be solved by considering different cases. The critical points for these cases are where the expressions inside the absolute values become zero, i.e., at [tex]\( x = 3 \)[/tex] and [tex]\( x = 2 \)[/tex].
Since [tex]\( x-3 \)[/tex] and [tex]\( x-2 \)[/tex] are linear functions inside the absolute values, these points divide the number line into intervals. Specifically, we should analyze the intervals split by these points:
- Interval 1: [tex]\( x < 2 \)[/tex]
- Interval 2: [tex]\( 2 \le x < 3 \)[/tex]
- Interval 3: [tex]\( x \ge 3 \)[/tex]
3. Analyze Each Interval:
- For [tex]\( x < 2 \)[/tex]:
- [tex]\( |x-3| = 3 - x \)[/tex]
- [tex]\( |x-2| = 2 - x \)[/tex]
The inequality [tex]\( |x-3| < |x-2| \)[/tex] becomes:
[tex]\[ 3 - x < 2 - x \][/tex]
Simplifying this, we see:
[tex]\[ 3 - x < 2 - x \][/tex]
This simplifies to:
[tex]\[ 3 < 2 \][/tex]
which is obviously false. Therefore, there are no solutions in this interval.
- For [tex]\( 2 \le x < 3 \)[/tex]:
- [tex]\( |x-3| = 3 - x \)[/tex]
- [tex]\( |x-2| = x - 2 \)[/tex]
The inequality [tex]\( |x-3| < |x-2| \)[/tex] becomes:
[tex]\[ 3 - x < x - 2 \][/tex]
Simplifying this, we get:
[tex]\[ 3 - x < x - 2 \][/tex]
Adding [tex]\( x \)[/tex] to both sides:
[tex]\[ 3 < 2x - 2 \][/tex]
Adding 2 to both sides:
[tex]\[ 5 < 2x \][/tex]
Dividing both sides by 2:
[tex]\[ x > \frac{5}{2} \][/tex]
Therefore, the solution in this interval is:
[tex]\[ \frac{5}{2} < x < 3 \][/tex]
- For [tex]\( x \ge 3 \)[/tex]:
- [tex]\( |x-3| = x - 3 \)[/tex]
- [tex]\( |x-2| = x - 2 \)[/tex]
The inequality [tex]\( |x-3| < |x-2| \)[/tex] becomes:
[tex]\[ x - 3 < x - 2 \][/tex]
Simplifying this, we get:
[tex]\[ x - 3 < x - 2 \][/tex]
Subtracting [tex]\( x \)[/tex] from both sides:
[tex]\[ -3 < -2 \][/tex]
which is always true. Therefore, every [tex]\( x \)[/tex] in this interval is part of the solution set.
4. Combine the Intervals:
From the analysis, we see that the solutions come from two intervals:
- [tex]\( \frac{5}{2} < x < 3 \)[/tex]
- [tex]\( x \ge 3 \)[/tex]
When combined, these intervals give the solution:
[tex]\[ \frac{5}{2} < x < \infty \][/tex]
Therefore, the solution set for the inequality [tex]\( |x-3| < |x-2| \)[/tex] is:
[tex]\[ \boxed{\frac{5}{2} < x < \infty} \][/tex]
1. Understand the Inequality: The inequality involves absolute values, which means it looks at the distance of [tex]\( x \)[/tex] from 3 and [tex]\( x \)[/tex] from 2 and states that the distance from 3 is less than the distance from 2.
2. Consider Different Cases: Absolute value inequalities can be solved by considering different cases. The critical points for these cases are where the expressions inside the absolute values become zero, i.e., at [tex]\( x = 3 \)[/tex] and [tex]\( x = 2 \)[/tex].
Since [tex]\( x-3 \)[/tex] and [tex]\( x-2 \)[/tex] are linear functions inside the absolute values, these points divide the number line into intervals. Specifically, we should analyze the intervals split by these points:
- Interval 1: [tex]\( x < 2 \)[/tex]
- Interval 2: [tex]\( 2 \le x < 3 \)[/tex]
- Interval 3: [tex]\( x \ge 3 \)[/tex]
3. Analyze Each Interval:
- For [tex]\( x < 2 \)[/tex]:
- [tex]\( |x-3| = 3 - x \)[/tex]
- [tex]\( |x-2| = 2 - x \)[/tex]
The inequality [tex]\( |x-3| < |x-2| \)[/tex] becomes:
[tex]\[ 3 - x < 2 - x \][/tex]
Simplifying this, we see:
[tex]\[ 3 - x < 2 - x \][/tex]
This simplifies to:
[tex]\[ 3 < 2 \][/tex]
which is obviously false. Therefore, there are no solutions in this interval.
- For [tex]\( 2 \le x < 3 \)[/tex]:
- [tex]\( |x-3| = 3 - x \)[/tex]
- [tex]\( |x-2| = x - 2 \)[/tex]
The inequality [tex]\( |x-3| < |x-2| \)[/tex] becomes:
[tex]\[ 3 - x < x - 2 \][/tex]
Simplifying this, we get:
[tex]\[ 3 - x < x - 2 \][/tex]
Adding [tex]\( x \)[/tex] to both sides:
[tex]\[ 3 < 2x - 2 \][/tex]
Adding 2 to both sides:
[tex]\[ 5 < 2x \][/tex]
Dividing both sides by 2:
[tex]\[ x > \frac{5}{2} \][/tex]
Therefore, the solution in this interval is:
[tex]\[ \frac{5}{2} < x < 3 \][/tex]
- For [tex]\( x \ge 3 \)[/tex]:
- [tex]\( |x-3| = x - 3 \)[/tex]
- [tex]\( |x-2| = x - 2 \)[/tex]
The inequality [tex]\( |x-3| < |x-2| \)[/tex] becomes:
[tex]\[ x - 3 < x - 2 \][/tex]
Simplifying this, we get:
[tex]\[ x - 3 < x - 2 \][/tex]
Subtracting [tex]\( x \)[/tex] from both sides:
[tex]\[ -3 < -2 \][/tex]
which is always true. Therefore, every [tex]\( x \)[/tex] in this interval is part of the solution set.
4. Combine the Intervals:
From the analysis, we see that the solutions come from two intervals:
- [tex]\( \frac{5}{2} < x < 3 \)[/tex]
- [tex]\( x \ge 3 \)[/tex]
When combined, these intervals give the solution:
[tex]\[ \frac{5}{2} < x < \infty \][/tex]
Therefore, the solution set for the inequality [tex]\( |x-3| < |x-2| \)[/tex] is:
[tex]\[ \boxed{\frac{5}{2} < x < \infty} \][/tex]
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