Question

solve the inequality $n - \frac{3}{5} \leq -\frac{4}{7}$ for $n$.\
\bigcirc\\ $n \geq -\frac{41}{35}$\
\bigcirc\\ $n \leq -\frac{1}{35}$\
\bigcirc\\ $n \leq \frac{1}{35}$\
\bigcirc\\ $n \geq \frac{41}{35}$

Explanation:

Step1: Add $\frac{3}{5}$ to both sides

To solve for \( n \), we add \(\frac{3}{5}\) to both sides of the inequality \( n - \frac{3}{5} \leq -\frac{4}{7} \). This gives us \( n \leq -\frac{4}{7} + \frac{3}{5} \).

Step2: Find a common denominator

The common denominator of 7 and 5 is 35. So we rewrite the fractions: \( -\frac{4}{7} = -\frac{20}{35} \) and \( \frac{3}{5} = \frac{21}{35} \). Then the inequality becomes \( n \leq -\frac{20}{35} + \frac{21}{35} \).

Step3: Add the fractions

Adding the numerators: \( -\frac{20}{35} + \frac{21}{35} = \frac{-20 + 21}{35} = \frac{1}{35} \)? Wait, no, wait. Wait, \( -\frac{4}{7} + \frac{3}{5} \): let's recalculate. Wait, \( -\frac{4}{7} \) is \( -\frac{20}{35} \), \( \frac{3}{5} \) is \( \frac{21}{35} \). So \( -\frac{20}{35} + \frac{21}{35} = \frac{1}{35} \)? But wait, the original inequality is \( n - \frac{3}{5} \leq -\frac{4}{7} \), so adding \( \frac{3}{5} \) to both sides: \( n \leq -\frac{4}{7} + \frac{3}{5} \). Wait, no, I think I made a mistake. Wait, \( -\frac{4}{7} + \frac{3}{5} \): let's compute \( \frac{3}{5} - \frac{4}{7} \). The common denominator is 35. \( \frac{3}{5} = \frac{21}{35} \), \( \frac{4}{7} = \frac{20}{35} \). So \( \frac{21}{35} - \frac{20}{35} = \frac{1}{35} \). Wait, but \( -\frac{4}{7} + \frac{3}{5} = \frac{3}{5} - \frac{4}{7} = \frac{1}{35} \)? Wait, no, \( -\frac{4}{7} + \frac{3}{5} = \frac{3}{5} - \frac{4}{7} = \frac{21 - 20}{35} = \frac{1}{35} \). Wait, but let's check again. Wait, the original inequality is \( n - \frac{3}{5} \leq -\frac{4}{7} \). So adding \( \frac{3}{5} \) to both sides: \( n \leq -\frac{4}{7} + \frac{3}{5} \). Let's compute \( -\frac{4}{7} + \frac{3}{5} \):

\( -\frac{4}{7} + \frac{3}{5} = \frac{3}{5} - \frac{4}{7} = \frac{21 - 20}{35} = \frac{1}{35} \)? Wait, that can't be right. Wait, no, \( -\frac{4}{7} \) is negative, \( \frac{3}{5} \) is positive. So \( -\frac{4}{7} + \frac{3}{5} = \frac{3}{5} - \frac{4}{7} \). Let's compute the decimal values: \( \frac{3}{5} = 0.6 \), \( \frac{4}{7} \approx 0.571 \). So \( 0.6 - 0.571 \approx 0.02857 \), which is \( \frac{1}{35} \approx 0.02857 \). So that's correct. So \( n \leq \frac{1}{35} \)? Wait, but let's check the options. One of the options is \( n \leq \frac{1}{35} \). Wait, but let's re-express the original inequality. Wait, \( n - \frac{3}{5} \leq -\frac{4}{7} \). Let's add \( \frac{3}{5} \) to both sides: \( n \leq -\frac{4}{7} + \frac{3}{5} \). Let's compute \( -\frac{4}{7} + \frac{3}{5} \):

\( -\frac{4}{7} = -\frac{20}{35} \), \( \frac{3}{5} = \frac{21}{35} \). So \( -\frac{20}{35} + \frac{21}{35} = \frac{1}{35} \). So \( n \leq \frac{1}{35} \). Wait, but let's check the options. The third option is \( n \leq \frac{1}{35} \). So that's the answer. Wait, but let me check again. Wait, \( n - \frac{3}{5} \leq -\frac{4}{7} \). Add \( \frac{3}{5} \) to both sides: \( n \leq -\frac{4}{7} + \frac{3}{5} \). Compute \( -\frac{4}{7} + \frac{3}{5} \):

\( -\frac{4}{7} + \frac{3}{5} = \frac{3}{5} - \frac{4}{7} = \frac{21 - 20}{35} = \frac{1}{35} \). So \( n \leq \frac{1}{35} \). So the correct option is the third one: \( n \leq \frac{1}{35} \). Wait, but let me check the arithmetic again. \( -\frac{4}{7} + \frac{3}{5} \):

\( \frac{3}{5} = 0.6 \), \( -\frac{4}{7} \approx -0.571 \). So \( 0.6 + (-0.571) = 0.02857 \), which is \( \frac{1}{35} \approx 0.02857 \). So that's correct. So the solution is \( n \leq \frac{1}{35} \).

Answer:

\( n \leq \frac{1}{35} \) (the third option: \( n \leq \frac{1}{35} \))