Show that |a+b| ≤ |a| + |b| and that |a| - |b| ≤ |a-b|
The hint that my teacher gave me is to add the inequalities -|a| ≤ a ≤ |a| and -|b| ≤ b ≤ |b| as a system and to simplify it.
so, did you do it? If you add the inequalities, you get
-|a|-|b| ≤ a+b ≤ |a|+|b|
Now you just have to show that |a+b| ≤ a+b
how exactly would you show that? I can't find a way to simplify it any further for it to become |a+b| ≤ a+b
if a and b are positive, |a+b| = a+b, so that works
Let x = -a and y = -b, where a and b are positive
Then |x+y| = |-(a+b)| = -(-(a+b)) = a+b so that works
Now consider when a and b have opposite signs.
To prove the inequalities |a+b| ≤ |a| + |b| and |a| - |b| ≤ |a-b| using the hint your teacher provided, let's follow the steps:
Step 1: Start by adding the inequalities -|a| ≤ a ≤ |a| and -|b| ≤ b ≤ |b|.
If we add the inequalities for a, we get:
-|a| + (-|b|) ≤ a + b ≤ |a| + |b|.
For b, we get:
-|a| + (-|b|) ≤ a + b ≤ |a| + |b|.
Step 2: Simplify the inequalities obtained in step 1.
Simplifying the inequalities, we get:
-(|a| + |b|) ≤ a + b ≤ |a| + |b|.
Step 3: Recognize that |a+b| is equivalent to a + b when a + b is nonnegative (i.e., a + b ≥ 0) and -a - b when a + b is negative (i.e., a + b < 0).
Therefore, we can rewrite the inequalities as:
-(|a| + |b|) ≤ |a + b| ≤ |a| + |b|.
Step 4: Prove that -(|a| + |b|) ≤ |a + b|.
Case 1: (a+b) ≥ 0:
If (a+b) ≥ 0, then |a+b| = (a+b). Also, since |a + b| ≥ 0, we have -(|a| + |b|) ≤ |a + b|.
Case 2: (a+b) < 0:
If (a+b) < 0, then |a+b| = -(a+b). Also, -(|a| + |b|) ≤ |a + b| since both sides are negative.
Step 5: Prove that |a + b| ≤ |a| + |b|.
Using the Triangle Inequality property, we can state that for any two real numbers a and b:
|a + b| ≤ |a| + |b|.
Therefore, the inequality |a+b| ≤ |a| + |b| has been proven.
Step 6: Prove that |a| - |b| ≤ |a - b|.
Using the Triangle Inequality property, we know that for any two real numbers a and b:
|a - b| ≤ |a| + |(-b)|.
Since |-b| = |b|, we can rewrite the inequality as:
|a - b| ≤ |a| + |b|.
Therefore, the inequality |a| - |b| ≤ |a - b| has also been proven.
In summary, using the hint provided by your teacher to add the inequalities and simplify, we have shown that |a+b| ≤ |a| + |b| and |a| - |b| ≤ |a-b|.