Question

Manuel takes a job translating English instruction manuals to Spanish. He will receive $15 per page plus $100 per month. Manuel would like to work for 3 months during the summer and make at least $1500. Please help explain how to write and solve an inequality to find the minimum number of pages Manuel must translate in order to reach his goal.

Accepted Answer

Let's review the information we know and then set up an inequality.  We know that Manuel wants to earn a minimum of $1500 over a 3 month period.  He earns $15 per translated page and we are trying to determine the number of pages he must translate, so let's call the number of pages he must translate x.  If we multiply the two, then the total amount he earns from pure translating is 15x.  But he also earns an extra $100 per month, so over three months that is $300 that must also be taken into account (or added to 15x).  So the inequality to set up is shown below.
15x + 300 gt= 1500 
Subtract 300 from both sides.
15x gt= 1200 
Now divide both sides by 15.
x gt= 80 
So he must translate at least 80 pages to make at least $1500.

Answer

So Manuel receives $15 per page and $100 a month. He would like to work for three months. The inequality that can be set up here is:
15a + 100m ≥ 1500'a' = the amount of pages translated, 'm' = the number of months he works. The '≥' symbol is used because he wants to make at least $1500. That means he can make greater than $1500 or just $1500 and it will still meet his goal. 
If he works for 3 months, and 'm' = the number of months he works, 3 can be substituted for 'm'. The updated inequality:
15a + 100(3) ≥ 1500. And then you simplify:
15a + 300 ≥ 1500. Then you start solving it like an equation:
15a + 300 - 300 ≥ 1500 - 300. Subtract 300 from both sides, first.
 15/15 a >= 1200/15 Then divide by 15 to isolate 'a'
a >= 80 
So according to the solution above, Manuel must translate at least 80 pages to make at least $1500

Math

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