Solve the inequality (x - 1)/2 < (7x + 2)/5 and give the solution in interval notation.

Step-by-step solution

1. 1

   Multiply by the LCD, and check its sign first. The denominators $2$ and $5$ give an LCD of $10$. Because $10>0$, multiplying both sides leaves the inequality direction unchanged — the direction only reverses when you multiply or divide by a negative number:

   $10\cdot\frac{x-1}{2}<10\cdot\frac{7x+2}{5}\;\Longrightarrow\;5(x-1)<2(7x+2).$

2. 2

   Expand both products. Distributing carefully:

   $5(x-1)=5x-5,\qquad 2(7x+2)=14x+4,$

   so the inequality reads

   $5x-5<14x+4.$

   Cross-multiplying "diagonally" without writing the LCD step is where students often pair the wrong numerator with the wrong denominator; the explicit multiply-by-10 line prevents that.

3. 3

   Move the $x$ terms to the side with the larger coefficient. Subtracting $5x$ from both sides keeps the coefficient of $x$ positive:

   $-5<9x+4.$

   The alternative — subtracting $14x$ — leaves $-9x-5<4$ and forces a division by $-9$, with a sign flip to remember. Choosing the direction that avoids a negative coefficient removes an entire class of error.

4. 4

   Isolate $x$. Subtract $4$, then divide by the positive number $9$:

   $-9<9x\;\Longrightarrow\;-1<x,\qquad\text{that is}\qquad x>-1.$

   In interval notation the solution set is $(-1,\infty)$; the endpoint is excluded because the original inequality is strict.

5. 5

   Verify around the boundary. At $x=-0.9$: left $=\frac{-1.9}{2}=-0.95$, right $=\frac{-4.3}{5}=-0.86$, and $-0.95<-0.86$ ✓. At $x=-1.1$: left $=-1.05$, right $=\frac{-5.7}{5}=-1.14$, so the inequality fails ✓. At $x=-1$ both sides equal $-1$, confirming the boundary.