Question

this question has two parts. first, answer part a. then, answer part b.
part a
consider the inequality (\frac{x}{10} + \frac{1}{4} geq \frac{1}{5}).
solve the inequality.

Explanation:

Step1: Subtract $\frac{1}{4}$ from both sides

To isolate the term with \( x \), we subtract \(\frac{1}{4}\) from both sides of the inequality \(\frac{x}{10}+\frac{1}{4}\geq\frac{1}{5}\). This gives us \(\frac{x}{10}\geq\frac{1}{5}-\frac{1}{4}\).

Step2: Find a common denominator and subtract

The common denominator of 5 and 4 is 20. So, \(\frac{1}{5}=\frac{4}{20}\) and \(\frac{1}{4}=\frac{5}{20}\). Then \(\frac{1}{5}-\frac{1}{4}=\frac{4}{20}-\frac{5}{20}=-\frac{1}{20}\). So the inequality becomes \(\frac{x}{10}\geq-\frac{1}{20}\).

Step3: Multiply both sides by 10

To solve for \( x \), we multiply both sides of the inequality \(\frac{x}{10}\geq-\frac{1}{20}\) by 10. This gives \( x\geq-\frac{1}{20}\times10 \).

Step4: Simplify the right - hand side

\(-\frac{1}{20}\times10 = -\frac{10}{20}=-\frac{1}{2}\). So \( x\geq-\frac{1}{2}\).

Answer:

\(x\geq-\frac{1}{2}\)