To solve the equation, we can start by simplifying both sides:

11/13 - (5/42 - x) = -(15/28 - 11/13)

To simplify the left side, we need to subtract (5/42 - x):

11/13 - 5/42 + x = -(15/28 - 11/13)

To simplify the right side, we need to subtract (15/28 - 11/13):

11/13 - 5/42 + x = -[(15/28) - (11/13)]

Now, let's simplify the fractions:

11/13 - 5/42 + x = -[(165/364) - (308/364)]

11/13 - 5/42 + x = -[-143/364]

11/13 - 5/42 + x = 143/364

Now, let's find a common denominator for the fractions on the left side:

11/13 - (5/42)(13/13) + x = 143/364

11/13 - 65/546 + x = 143/364

Now, let's simplify the fractions:

(11*42 - 65*13)/(13*42) + x = 143/364

(462 - 845)/(546) + x = 143/364

(-383)/(546) + x = 143/364

Now, let's simplify the fraction on the left side:

-383/546 + x = 143/364

To get rid of the fractions, we can multiply both sides by the least common multiple (LCM) of the denominators, which is 364*546:

(364*546)(-383/546) + (364*546)(x) = (364*546)(143/364)

-383(364) + 364(546)x = 143(546)

Now, let's simplify the equation:

-139732 + 198744x = 77978

To isolate x, we can subtract -139732 from both sides:

-139732 + 198744x - (-139732) = 77978 - (-139732)

198744x = 217710

Now, let's solve for x by dividing both sides by 198744:

x = 217710/198744

x ≈ 1.095