Solve the inequality 2x - 1/5 < x + 1/2.

Step-by-step solution

1. 1

   Decide whether to clear denominators or move terms first. Multiplying everything by $10$ would work, but here the variable terms are simple, so it is quicker to separate variables from constants and combine the two fractions only once.

2. 2

   Move the $x$ terms left and the constants right. Subtract $x$ from both sides and add $\tfrac15$:

   $2x-x<\frac{1}{2}+\frac{1}{5}$

   $x<\frac{1}{2}+\frac{1}{5}$

3. 3

   Add the fractions over the least common denominator. The LCD of $2$ and $5$ is $10$:

   $\frac{1}{2}=\frac{5}{10},\qquad\frac{1}{5}=\frac{2}{10}$

   $\frac{5}{10}+\frac{2}{10}=\frac{7}{10}$

4. 4

   State the solution. The coefficient of $x$ is $+1$, so nothing needs dividing and no sign flips:

   $x<\frac{7}{10}=0.7$

5. 5

   Check either side of the boundary. At $x=0$: left $=-0.2$, right $=0.5$, and $-0.2<0.5$ $\checkmark$. At $x=1$: left $=1.8$, right $=1.5$, false $\checkmark$. At $x=0.7$ both sides equal $1.2$, so the strict inequality correctly excludes it.