Question

mustafa, heloise, and gia have written more than a combined total of 22 articles for the school newspaper. heloise has written \\(\frac{1}{4}\\) as many articles as mustafa has. gia has written \\(\frac{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? choose 1 answer: a \\(m > \frac{1}{2}\\) b \\(m geq \frac{1}{2}\\) c \\(m > 8\\) d \\(m geq 8\\)

Explanation:

Part 1: Writing the Inequality

Step 1: Define the number of articles each wrote

Let \( m \) be the number of articles Mustafa wrote. Heloise wrote \( \frac{1}{4}m \) articles, and Gia wrote \( \frac{3}{2}m \) articles.

Step 2: Set up the inequality for total articles

The combined total of their articles is more than 22, so \( m + \frac{1}{4}m + \frac{3}{2}m > 22 \).

Step 3: Combine like terms

First, find a common denominator (4) for the terms: \( m = \frac{4}{4}m \), \( \frac{3}{2}m = \frac{6}{4}m \). So, \( \frac{4}{4}m + \frac{1}{4}m + \frac{6}{4}m > 22 \), which simplifies to \( \frac{4 + 1 + 6}{4}m > 22 \), or \( \frac{11}{4}m > 22 \).

Step 1: Solve \( \frac{11}{4}m > 22 \)

Multiply both sides by \( \frac{4}{11} \) (the reciprocal of \( \frac{11}{4} \)) to isolate \( m \).

Step 2: Calculate the result

\( m > 22 \times \frac{4}{11} \), and \( 22 \times \frac{4}{11} = 8 \), so \( m > 8 \).

Answer:

The inequality is \( m + \frac{1}{4}m + \frac{3}{2}m > 22 \) (or simplified as \( \frac{11}{4}m > 22 \)).

Part 2: Solving the Inequality