Questions: Solve the inequality for (u). [ frac37 u+1>frac97-frac45 u ] Simplify your answer as much as possible.

Solution Steps

To solve the inequality for \( u \), we need to isolate \( u \) on one side of the inequality. This involves combining like terms and performing basic algebraic operations such as addition, subtraction, multiplication, and division.

We start with the inequality: \[ \frac{3}{7} u + 1 > \frac{9}{7} - \frac{4}{5} u \]

To isolate \( u \), we first move all terms involving \( u \) to one side and constant terms to the other side. This gives us: \[ \frac{3}{7} u + \frac{4}{5} u > \frac{9}{7} - 1 \]

Calculating the right side: \[ \frac{9}{7} - 1 = \frac{9}{7} - \frac{7}{7} = \frac{2}{7} \] Now, we combine the \( u \) terms on the left side: \[ \left(\frac{3}{7} + \frac{4}{5}\right) u > \frac{2}{7} \] Finding a common denominator (35) for the coefficients of \( u \): \[ \frac{15}{35} + \frac{28}{35} = \frac{43}{35} \] Thus, we have: \[ \frac{43}{35} u > \frac{2}{7} \]

To isolate \( u \), we multiply both sides by \( \frac{35}{43} \): \[ u > \frac{2}{7} \cdot \frac{35}{43} = \frac{70}{301} \] This simplifies to: \[ u > 0.2326 \]

Final Answer