Find all numbers x for which |x-1| + |x-2| > 1.
I know the answer is that there exists no such x, but I need someone to explain to me the steps as to figuring out why that is the case.
Between 1 and 2, you have
y = (x-1) + -(x-2) = 1
everywhere else, y > 1
for x < 1, y = -(x-1) + -(x-2) = 3-2x
for x > 2, y = (x-1) + (x-2) = 2x-3
| x - 1 | =
x - 1
OR
- | x - 1 | = 1 - x
| x - 2 | =
x - 2
OR
- | x - 2 | = 2 - x
4 combinations are possible.
1. combination:
x - 1 + x - 2 > 1
2 x - 3 > 1
2 x > 1 + 3
2 x > 4 Divide both sides by 2
x > 2
2. combination:
x - 1 + 2 - x > 1
1 > 1
That is not true and that is not solution.
3. combination:
1 - x + x - 2 > 1
- 1 > 1
Again, that is not true and that is not solution.
4. combination:
1 - x + 2 - x > 1
- 2 x + 3 > 1
- 2 x > 1 - 3
- 2 x > - 2 Divide both sides by - 2
When you multiply or divide both sides of inequality by a negative number, inequality change the direction. So:
x < 1
Solutions are :
x < 1
and
x > 2
To solve the inequality |x-1| + |x-2| > 1, we can break it down into several cases and analyze each case separately.
Case 1: x < 1
In this case, both expressions |x-1| and |x-2| will be negative. Therefore, their sum will always be negative, which means the inequality |x-1| + |x-2| > 1 cannot hold true for any values of x less than 1.
Case 2: 1 ≤ x < 2
In this case, |x-1| will be non-negative, but |x-2| will still be negative. The sum of a non-negative and a negative number will always be negative. Therefore, the inequality |x-1| + |x-2| > 1 cannot hold true for any values of x in the range 1 ≤ x < 2.
Case 3: 2 ≤ x
In this case, both expressions |x-1| and |x-2| will be non-negative. The sum of two non-negative numbers can be greater than 1. However, |x-1| + |x-2| will always be less than or equal to 2 because the largest possible value for |x-1| + |x-2| is achieved when x is equal to 2, which gives |2-1| + |2-2| = 1 + 0 = 1. Therefore, the inequality |x-1| + |x-2| > 1 cannot hold true for any values of x greater than or equal to 2.
Combining all three cases, we can conclude that there are no values of x for which the inequality |x-1| + |x-2| > 1 is satisfied.