Solve the inequality (2x - x)/3 < (1/3)(9 - 5x).

Step-by-step solution

1. 1

   Simplify the numerator. The like terms combine as $2x-x=x$, so the inequality reads $\frac{x}{3}<\frac13(9-5x).$

2. 2

   Clear the common denominator. Multiplying both sides by $3$, a positive number, keeps the direction of the inequality: $x<9-5x .$

3. 3

   Gather the x terms. Adding $5x$ to both sides gives $6x<9 .$

4. 4

   Divide and reduce the fraction. Dividing by the positive number $6$ gives $x<\frac96$, and reducing by $3$ gives $x<\frac32 .$

5. 5

   Check the boundary and one interior point. At $x=\frac32$ both sides equal $\frac12$, so the strict inequality fails exactly at the boundary, as expected. At $x=0$ the left side is $0$ and the right side is $3$, so the inequality holds; the solution set is $\left(-\infty,\frac32\right)$.