Question 1014611: write an inequality to this problem and solve: Hunter earns twice as much per week as his sister Naomi. if she makes $30 per week more than her brother Jake and the sum of their salaries is at most $330, what range can their salaries take each week?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! here's what i think.
h = money hunter makes in a week.
n = money naomi makes in a week.
j = money jake makes in a week.
h + n + j <= 330
this means that the total that they all make in a week has to be less than or equal to 330.
you are given that hunter makes two times as much in a week as naomi.
the equation for this would be h = 2n.
you are given that naomi makes 30 more in a week than jake.
the equation for this is n = j + 30
you have 3 equations to work with.
h + n + j <= 330
h = 2n
n = j + 30
you want to convert everything in terms of one of the variables if that is possible.
since h = 2n and n = j + 30, then h = 2n becomes h = 2 * (j + 30) which becomes h = 2j + 60.
your equation are now:
h + n + j <= 330
h = 2j + 60
n = j + 30
in the first equation, you can replace n with j + 30 and you can replace h with 2j + 60 to get:
h + n + j <= 330 becomes (2j + 60) + (j + 30) + j <= 330.
remove parentheses to get 2j + 60 + j + 30 + j <= 330.
combine like terms to get 4j + 90 <= 330.
subtract 90 from both sides of this equation to get 4j <= 240.
divide both sides of this equation by 4 to get j <= 60.
since n = j + 30, then if j <= 60, n has to be <= 90
since h = 2j + 60, then if j <= 60, h has to be <= 180.
you get:
j <= 60
n <= 90
h <= 180
the maximum they can be is:
j = 60
n = 90
h = 180
the total of j + n + h would be 330.
this agrees with the problem statement so far.
the minimum they can be is calculated as follows:
the smallest one is j.
none of them can go negative.
the smallest j can be is 0.
the smallest n can be is 30 because n = j + 30.
when j is equal to 0, n is equal to 30.
if n went below 30, j would be negative, which is not allowed.
the smallest h can be is 60 because h = 2j + 60.
when j is equal to 0, h is equal to 60.
if h went below 60, j would be negative, which is not allowed.
hunters range is from 60 to 180 inclusive.
naomi's range is from 30 to 90 inclusive.
jake's range is from 0 to 60 inclusive.
in interval notation, you get:
h = [60,180]
n = [30,90]
j = [0,60]
based on this, the minimum they all can be is 90 total and the maximum they all can be is 330 total.
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