Let a and b be the maximum and minimum value of x, respectively, satisfying the inequality |x+19|+|x−22|≤137, then what is a−b?
Clearly for -19 <= x <= 22 the sum is constant at 41.
On each end of the interval, the sum rises by 2 for each change of 1 in x.
Since 137-41 = 96, divide that by 2 to see that as long as x is within 48 of the endpoints of [-19,22] the sum is within 96 of 41.
So, -67 <= x <= 70
Well, I have to say, these mathematical expressions can sometimes make my circuits spin! But, fear not, I'll give it my best shot!
Let's break it down step by step:
First, let's consider the inner absolute values. Since we're looking for the maximum and minimum values of x, we can consider two cases:
Case 1: (x + 19) and (x - 22) are both non-negative.
In this case, both absolute values can be simplified to (x + 19) + (x - 22), which gives us: 2x - 3.
Case 2: (x + 19) is non-negative and (x - 22) is negative.
In this case, the absolute value (x + 19) simplifies to (x + 19) and the other absolute value (x - 22) simplifies to -(x - 22), which gives us: (x + 19) - (x - 22), which simplifies to: 41.
Now, let's combine both cases:
Case 1: 2x - 3 ≤ 137
We can solve this inequality and find: x ≤ 70.
Case 2: 41 ≤ 137
This inequality is satisfied for all values of x.
So, the minimum value of x is -∞ and the maximum value of x is 70.
Now, a - b = 70 - (-∞) = ∞.
Voila! The difference between a and b is infinity. But, don't worry, infinity is just a friendly mathematical term, not to be taken literally!
Remember, take my answers with a grain of circuitry, and embrace the humor in every equation!
To find the maximum and minimum values of x satisfying the inequality |x + 19| + |x - 22| ≤ 137, let's solve it step-by-step:
Step 1: Consider the two cases when the absolute values are positive and when they are negative.
Case 1: x + 19 is positive and x - 22 is positive.
When x + 19 is positive, we have x + 19 ≥ 0, which gives us x ≥ -19.
When x - 22 is positive, we have x - 22 ≥ 0, which gives us x ≥ 22.
The common region for these values is x ≥ 22.
Case 2: x + 19 is negative and x - 22 is negative.
When x + 19 is negative, we have x + 19 ≤ 0, which gives us x ≤ -19.
When x - 22 is negative, we have x - 22 ≤ 0, which gives us x ≤ 22.
The common region for these values is x ≤ -19.
Step 2: Combine the common regions from both cases.
The common regions for x are -19 ≤ x ≤ 22.
Step 3: Find the maximum and minimum values of x in the common region.
The maximum value of x is a = 22.
The minimum value of x is b = -19.
Step 4: Calculate a - b.
a - b = 22 - (-19) = 22 + 19 = 41.
Therefore, a - b is equal to 41.
To solve the inequality |x+19|+|x−22|≤137, we need to consider different cases based on the signs of (x+19) and (x-22).
Case 1: x ≤ -19
If x ≤ -19, then both (x+19) and (x-22) are negative. So, the inequality simplifies to (-x-19) + (-x+22) ≤ 137.
Simplifying further, we get -2x + 3 ≤ 137.
Subtracting 3 from both sides, we have -2x ≤ 134.
Dividing both sides by -2 (remember that we need to flip the inequality sign when dividing by a negative number), we get x ≥ -67.
Case 2: -19 ≤ x ≤ 22
If -19 ≤ x ≤ 22, then (x+19) is non-negative and (x-22) is negative. So, the inequality simplifies to (x+19) + (-x+22) ≤ 137.
Simplifying further, we get x - x + 19 + 22 ≤ 137.
Combining like terms, we have 41 ≤ 137.
Case 3: x ≥ 22
If x ≥ 22, then both (x+19) and (x-22) are non-negative. So, the inequality simplifies to (x+19) + (x-22) ≤ 137.
Simplifying further, we get 2x - 3 ≤ 137.
Adding 3 to both sides, we have 2x ≤ 140.
Dividing both sides by 2, we get x ≤ 70.
Therefore, the valid range for x is -67 ≤ x ≤ 70. The maximum value of x is 70 (a) and the minimum value of x is -67 (b).
To find a - b, we subtract the minimum value from the maximum value:
a - b = 70 - (-67) = 70 + 67 = 137.