Question

solve the inequality for w.

w - \frac{3}{4} < -\frac{3}{8}

simplify your answer as much as

Explanation:

Step1: Add $\frac{3}{4}$ to both sides

To isolate \( w \), we add \(\frac{3}{4}\) to both sides of the inequality \( w - \frac{3}{4} < -\frac{3}{8} \). This gives us \( w - \frac{3}{4} + \frac{3}{4} < -\frac{3}{8} + \frac{3}{4} \).

Step2: Simplify the right - hand side

First, find a common denominator for \(-\frac{3}{8}\) and \(\frac{3}{4}\). The common denominator of 8 and 4 is 8. So, \(\frac{3}{4}=\frac{3\times2}{4\times2}=\frac{6}{8}\). Then, \(-\frac{3}{8}+\frac{6}{8}=\frac{- 3 + 6}{8}=\frac{3}{8}\)? Wait, no, wait. Wait, the original inequality is \(w-\frac{3}{4}<-\frac{3}{8}\). When we add \(\frac{3}{4}\) to both sides:

\(w<-\frac{3}{8}+\frac{3}{4}\)

Convert \(\frac{3}{4}\) to eighths: \(\frac{3}{4}=\frac{6}{8}\)

So, \(-\frac{3}{8}+\frac{6}{8}=\frac{-3 + 6}{8}=\frac{3}{8}\)? No, wait, no. Wait, let's do it again.

Wait, \(w-\frac{3}{4}<-\frac{3}{8}\)

Add \(\frac{3}{4}\) to both sides:

\(w<-\frac{3}{8}+\frac{3}{4}\)

\(\frac{3}{4}=\frac{6}{8}\), so \(-\frac{3}{8}+\frac{6}{8}=\frac{3}{8}\)? Wait, no, that can't be. Wait, no, the correct calculation: \(-\frac{3}{8}+\frac{3}{4}=-\frac{3}{8}+\frac{6}{8}=\frac{3}{8}\)? Wait, no, \(-3 + 6 = 3\), so \(\frac{3}{8}\). Wait, but let's check with the original inequality. If \(w = \frac{3}{8}\), then \(w-\frac{3}{4}=\frac{3}{8}-\frac{6}{8}=-\frac{3}{8}\), which is equal to the right - hand side, but our inequality is strict (\(<\)), so \(w<\frac{3}{8}\)? Wait, no, wait, I think I made a mistake. Wait, the original inequality is \(w-\frac{3}{4}<-\frac{3}{8}\)

Let's solve it again:

\(w-\frac{3}{4}<-\frac{3}{8}\)

Add \(\frac{3}{4}\) to both sides:

\(w<-\frac{3}{8}+\frac{3}{4}\)

\(\frac{3}{4}=\frac{6}{8}\), so \(-\frac{3}{8}+\frac{6}{8}=\frac{3}{8}\). Wait, but if we take \(w = 0\), then \(0-\frac{3}{4}=-\frac{3}{4}\), and \(-\frac{3}{4}=-\frac{6}{8}\), and \(-\frac{6}{8}<-\frac{3}{8}\), which is true. And \(0<\frac{3}{8}\) is also true. If we take \(w=\frac{3}{8}\), then \(w - \frac{3}{4}=\frac{3}{8}-\frac{6}{8}=-\frac{3}{8}\), which is not less than \(-\frac{3}{8}\), so the solution is \(w<\frac{3}{8}\)

Wait, no, wait, I think I messed up the sign. Wait, \(-\frac{3}{8}+\frac{3}{4}\): \(\frac{3}{4}\) is positive, so adding a positive to \(-\frac{3}{8}\) will increase the value. So \(-\frac{3}{8}+\frac{3}{4}=\frac{-3 + 6}{8}=\frac{3}{8}\). So the solution is \(w<\frac{3}{8}\)

Wait, let's verify with \(w = 0\): \(0-\frac{3}{4}=-\frac{3}{4}\), and \(-\frac{3}{4}<-\frac{3}{8}\) (since \(-\frac{3}{4}=-\frac{6}{8}\) and \(-\frac{6}{8}<-\frac{3}{8}\)). If \(w=\frac{3}{8}\), then \(w-\frac{3}{4}=\frac{3}{8}-\frac{6}{8}=-\frac{3}{8}\), which is not less than \(-\frac{3}{8}\), so the solution is correct.

Answer:

\(w<\frac{3}{8}\)