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Thread: solution that satisfies inequalities

  1. #1
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    solution that satisfies inequalities

    I'm interested in the condition for existence of a solution $\displaystyle (x_1, x_2)$ given

    $\displaystyle |a-x_1|\leq C_1$
    $\displaystyle |x_1-x_2|\leq C_2$
    $\displaystyle |b-x_2|\leq C_3$
    $\displaystyle x_1,x_2\leq C_4$


    This is my conclusion. Let

    $\displaystyle I_1 = [a-C_1, a+C_1]$
    $\displaystyle I_2 = [b-C_3, b+C_3]$

    Now, $\displaystyle x_2 \in [x_1-C_2, x_1+C_2]$ and $\displaystyle x_1 \in [x_2-C_2,x_2+C_2]$

    So, if

    $\displaystyle I_3 = [a-C_1-C_2, a+C_1+C_2]$
    $\displaystyle I_4 = [b-C_3-C_2, b+C_3+C_2]$

    Then a solution exist if $\displaystyle I_A = I_1 \cap I_3 \cap [0, C_4] \neq \emptyset$ and $\displaystyle I_B = I_2 \cap I_4 \cap [0, C_4] \neq \emptyset$ and $\displaystyle x_1 \in I_A, x_2 \in I_B$


    Is this correct? Does anyone see a problem? Thanks
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  2. #2
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    Re: solution that satisfies inequalities

    I'm sorry for the rushed post. I meant to write that I'm looking for a pair of numbers $\displaystyle (x_1,x_2)$ that satisfies the first four inequalities listed in original post.

    If such a pair of numbers exist then I expect $\displaystyle x_1$ to one of possibly many values in some interval $\displaystyle I_A$. Similarly, $\displaystyle x_2$ could be one of many values in some interval $\displaystyle I_B$ depending on the values of the constants $\displaystyle a, b, C_1, ..., C_4$

    One mistake I made in my search for those intervals was mixing up $\displaystyle I_3$ and $\displaystyle I_4$.

    It seems that $\displaystyle x_1 \in [a-C_1,a+C_1] = I_1$ and $\displaystyle x_1 \in [b-C_3-C_2, b+C_3+C_2] = I_3$ and also $\displaystyle x_2 \in [b-C_3, b+C_3] = I_2$ and $\displaystyle x_2 \in [a-C_1-C_2,a+C_1+C_2] = I_4$ and so the intervals I'm looking for are just the intersections of these but I'm not totally sure
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