Linear Inequalities with Fractions Score: 0/1 Penalty: none Question Watch Video Show Examples Solve the following inequality: \[ x+\frac{3}{2}<\frac{1}{4}(-6 x-2)-\frac{3}{4} \] Answer \( \square \) Submit Answer \( \phi \) or \( \cup \) \( \infty \) [,] [,) (, ]

To solve the inequality \(x + \frac{3}{2} < \frac{1}{4}(-6x - 2) - \frac{3}{4}\), let’s first simplify the right side. Distributing the \(\frac{1}{4}\) gives us \(-\frac{6x}{4} - \frac{2}{4}\), which simplifies to \(-\frac{3}{2} - \frac{1}{2} = -2\). Now we rewrite the inequality: \[ x + \frac{3}{2} < -2. \] Next, subtract \(\frac{3}{2}\) from both sides: \[ x < -2 - \frac{3}{2}. \] Converting \(-2\) into halves gives us \(-\frac{4}{2}\), thus: \[ x < -\frac{4}{2} - \frac{3}{2} = -\frac{7}{2}. \] So the solution is \(x < -\frac{7}{2}\). For a fun side note, did you know that inequalities can often represent real-life situations, like comparing temperatures, prices, or distances? They help us understand limits and constraints in everyday decisions! If you want to dive deeper into solving inequalities, check out online platforms or math workbooks that specifically focus on linear equations and inequalities. They often have interactive exercises and practical applications to keep your learning engaging and exciting!