Solve the inequality (0.03y + 0.01)/0.03 - 1 < (0.7y - 0.3)/0.5 + 0.2(0.1y - 0.2)/0.15 and give the answer in the form y > ___.

Step-by-step solution

1. 1

   Split the left quotient term by term. A decimal denominator is easiest to remove one numerator term at a time: $\frac{0.03y+0.01}{0.03}=\frac{0.03y}{0.03}+\frac{0.01}{0.03}=y+\frac13$. Subtracting the $1$ gives a left side of $y-\frac23$.

2. 2

   Clear the 0.5 denominator by doubling. Dividing by $0.5$ is multiplying by $2$, so $\frac{0.7y-0.3}{0.5}=2(0.7y-0.3)=1.4y-0.6$. Recognising the reciprocal is faster and safer than long division with decimals.

3. 3

   Simplify the third fraction. First expand the numerator: $0.2(0.1y-0.2)=0.02y-0.04$. Then $\frac{0.02y-0.04}{0.15}=\frac{2y-4}{15}=\frac{2}{15}y-\frac{4}{15}$, obtained by multiplying numerator and denominator by $100$.

4. 4

   Add the two right-hand pieces over a common denominator of 15. Writing $1.4=\frac{21}{15}$ and $0.6=\frac{9}{15}$ gives $\frac{21}{15}y-\frac{9}{15}+\frac{2}{15}y-\frac{4}{15}=\frac{23}{15}y-\frac{13}{15}$.

5. 5

   Multiply the whole inequality by 15. The inequality is now $y-\frac23<\frac{23}{15}y-\frac{13}{15}$. Multiplying by the positive number $15$ keeps the direction of the inequality: $15y-10<23y-13$.

6. 6

   Isolate y. Subtract $15y$ and add $13$ to both sides: $3<8y$, so $y>\frac38$. Dividing by the positive $8$ again leaves the direction unchanged.

7. 7

   Check the boundary. At $y=\frac38$ both sides equal $-\frac{7}{24}$, so the endpoint is excluded. Testing $y=0.385$ gives left $=-0.2817$ and right $=-0.2763$, and $-0.2817<-0.2763$ is true, confirming the solution set.