Question 1121579
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Your equation, *[tex \Large m\ =\ 2b\ +\ 5] represents the <b>youngest</b> that Mary could be in relation to her brother's actual age.  But you are given that Mary is <i>at least</i> five years older than twice her brother's age, so an inequality is required to correctly model the situation.  So if the "equals" relationship is the youngest she can be, then she must either be equal to that age or <i>greater</i>, which is to say:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m\ \geq\ 2b\ +\ 5]


So what is *[tex \Large b] if *[tex \Large m\ =\ 7]?


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 7\ \geq\ 2b\ +\ 5]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2\ \geq\ 2b]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ b\ \leq\ 1]


And it certainly is possible for the brother to be one year old, and therefore it is, indeed, possible for Mary to be 7.  In fact, if you don't restrict the domain of your inequality to the integers, Mary could be a good deal younger than 7 years old.  Consider the situation where the brother was born 5 minutes ago.  Then Mary could be as young as 5 years and 10 minutes old.  Of course, you have no real idea about the upper bound of her age range except that there are practical limitations on the difference in ages of a given woman's oldest and youngest child.  For Mary to be 30 years older than her brother in the real world would be quite a stretch for their mother.


As to Mary's age when the brother is 10, just plug 10 into your inequality and do the indicated arithmetic:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m\ \geq\ 2(10)\ +\ 5]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m\ \geq\ 25]


*[illustration maryandbrother.jpg]
								
								
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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