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Question

Mustafa, Heloise, and Gia have written more than a combined total of 22 articles for the school newspaper. Heloise has written 1/4 as many articles as Mustafa has. Gia has written 3/2  as many articles as Mustafa has.

Write an inequality to determine the number of articles, m, Mustafa could have written for the school newspaper.

What is the solution set of the inequality?

Answer 1

Let h represent the articles written by Heloise, g represent the articles written by Gia, and m represent the articles written by Mustafa, then:

h = (1/4)m

g = (3/2)m

Because Heloise has written 1/4 as many articles as Mustafa has so we can multiply m, the articles written by Mustafa, by 1/4 to find the articles Heloise has written and Gia has written 3/2 as many articles as Mustafa has so we can multiply m, the articles written by Mustafa, by 3/2 to find the articles Gia has written.

If we add the number of articles each author has written together, we can determine the total number of articles written.

m + h + g > 22

We use m to represent the articles written by Mustafa, h to represent the articles written by Heloise, and g to represent the articles written by Gia. This quantity is greater than 22 as defined by the original question.

We can plug our identities, which we defined above, into our inequality.

m + (1/4)m + (3/2)m > 22

The leading term, m, can be expressed as (1/1)m because (1/1) = 1.

(1/1)m + (1/4)m + (3/2)m > 22

We now have three fractions to add together but each one has a different denominator than the other. The lowest common denominator between the three is four, so we will perform a little "mathemagic" such that each fraction will have the same denominator.

(4/4)(1/1)m + (1/4)m + (2/2)(3/2)m > 22

When we multiple a fraction by another fraction, we can multiply the numerators and the denominators together straight across. Notice that (4/4) = 1 and (2/2) = 1; therefore, we are not changing the quantity these fractions represent because multiplying by 1 does not change a number's quantity.

(4/4)m + (1/4)m + (6/4)m > 22

Now we can add the numerators of the fractions while leaving the denominators alone.

(11/4)m > 22

Ultimately we want to get m alone so we can find its value. We can start by multiplying both sides by 4.

(4)(11/4)m > (4)22

(44/4)m > 88

11m > 88

Our next step is to divide both sides by 11.

(11/11)m > (88/11)

m > 8

So m is greater than 8. We can represent this solution set as:

(8, ∞)

Where we use parenthesis, or "soft brackets," to represent that neither 8 nor the complex number, ∞, is included in the solution set.

I hope this has been helpful. Please do not hesitate to reach out with further questions!

Answer 2

Let's define some variables first:

M=Articles Mustafa has written

H=Articles Heloise has written

G= Articles Gia has written

We know that combined they all have written more than 22 articles. Therefore we can write the following formula:

M + H + G > 22 articles

Now, we also know that Heloise has written 1/4 times the number of articles Mustafa has written which means:

H = (1/4)*M. or H = M/4

We also know that Gia has written 3/2 times the number of articles Mustafa has written which means:

G = (3/2) * M or G=(3M)/2

We can now substitute the following equations into our inequality formula in order to eliminate all but one variable.

M + H + G > 22 turns into ->

M + M/4 + (3M)/2 > 22

From here we simply simplify and solve:

The greatest common denominator is 4, therefore we shall rewrite this formula as

(4M)/4 + M/4 + (6M)/4 > 22

Since we have now re-written it so all of the denominators are 4 we can add the numerators:

(4M + M + 6M)/4 > 22

(11M)/4 > 22

Now we solve for M

Multiply 4 to each side to eliminate the denominator on the left side

4*(11M/4) > 22 * 4 =

11M > 88

Now divide by 11 on each side

11M/11 > 88/11 =

M > 8

This means Mustafa has written at least 8 articles