(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

My question is whether the following inequality can be proven.

2. Relevant equations

[tex]

\left|\int_a^bg\left(x\right)dx-\int_a^bh\left(x\right)dx\right|\leq\int_a^b\left|g\left(x\right)-h\left(x\right)\right|dx

[/tex]

3. The attempt at a solution

I tried to write down the inequality in the form of it's primitives, where [tex]G\left(x\right)[/tex] is the primitive of [tex]g\left(x\right)[/tex] and [tex]H\left(x\right)[/tex] is the primitive of [tex]h\left(x\right)[/tex]. The inequality then becomes:

[tex]

\left|G\left(b\right)-G\left(a\right)-H\left(b\right)+H\left(a\right)\right|\leq\left|G\left(b\right)-H\left(b\right)\right|-\left|G\left(a\right)-H\left(a\right)\right|

[/tex]

But what next, or are there other means of getting a proof?

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# Is this inequality true and provable?

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