The discussion focuses on proving the properties of absolute value inequalities: |a+b|≤|a|+|b| and |a+b|≥|a|-|b| for various sign scenarios of a and b. It establishes that when both a and b are positive, the inequalities hold true as |a+b| equals |a|+|b|. For the case where both a and b are negative, the absolute values are expressed as |a| = -a and |b| = -b, leading to |a+b| = -a-b. Additionally, when a is positive and b is negative, the outcome depends on the relative magnitudes of |a| and |b|, confirming the inequalities under these conditions.
PREREQUISITESStudents studying algebra, mathematics educators, and anyone looking to deepen their understanding of absolute value inequalities and their applications in various sign scenarios.
If a and b are both less than 0, then so is a+ b.wonnabewith said:Homework Statement
I was trying to show that
1) |a+b|≤|a|+|b|
2) |a+b|≥|a|-|b|
and find out how they were true when a,b>0, a,b<0, and a>0,b<0
Homework Equations
1) |a+b|≤|a|+|b|
2) |a+b|≥|a|-|b|
The Attempt at a Solution
For |a+b|≤|a|+|b|
a,b>0
I got that |a+b|=a+b
|a|+|b|=a+b
so |a+b|=|a|+|b|
but I was confused when I was trying to solve a,b<0, and a>0,b<0
please help!