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This is false if you are aware of every word you wrote in this question! The increasing in the price is not related to how much a consumer spend on good X, it is related to the quantity the consumer buy of X. Say the increas in the price of X is 10 dollars, prior this increas the consumer used to spend 100 dollars on X, after increasing he/she could still soending 100 but the quantity he/she purchases is less.
In short, I want to say when goods are complements the increase in deman of X will lead to an increase in demand for Y, and we are not discussing how much the consumer spend on X or Y.
It is true.
If the price of X increases, the demand decreases (the willingness to pay decreases).Therefore, the demand of Y also decreases, as they are complementary.
If two goods X and Y are such that they are complements,an increase in the price of X will reduce the quantity demanded for it(law of demand). Since there is a reduction in the quantity of X,less of Y will also be demanded since they are complements.E.g let say the demand for good X is 20units,with 10units of good Y complementing it.if an increase in price of X leads to reduction in quantity demanded by 10unitsi.e new demand for X is 10units,then d demand of the complement Y is gonna reduce to 5units given 1unit of good Y complements 2units of good X.The relationship between the price of a good and d quantity of it complement is NEGATIVE OR INVERSE,while that the price of a good and the quantity of it substitute is POSITIVE OR DIRECT