Solve the inequality 3x + 1 - 300 < 4.2x - 1.8 + 4x - 80.

Step-by-step solution

1. 1

   Simplify each side separately before moving anything. Left: $3x+1-300 = 3x-299$. Right: the $x$-terms give $4.2x+4x = 8.2x$ and the constants give $-1.8-80 = -81.8$, so the right side is $8.2x-81.8$. The inequality is now $3x-299 < 8.2x-81.8.$

2. 2

   Move the x-terms to the side that keeps the coefficient positive. Subtracting $3x$ from both sides leaves $8.2x-3x = 5.2x$ on the right: $-299 < 5.2x-81.8.$ Choosing this direction avoids a later division by a negative number, which would flip the inequality sign.

3. 3

   Isolate the x-term. Add $81.8$ to both sides: $-299+81.8 = -217.2 < 5.2x.$

4. 4

   Divide by 5.2 and keep the sign. Because $5.2>0$ the direction of the inequality is unchanged: $x > \frac{-217.2}{5.2}.$

5. 5

   Evaluate the boundary exactly. Multiplying numerator and denominator by $10$ gives $\dfrac{-2172}{52} = -\dfrac{543}{13} = -41.\overline{769230}$, so $x > -\frac{543}{13} \approx -41.77.$ The endpoint is excluded because the original inequality is strict.

6. 6

   Test a value on each side. At $x=0$: left $=-299$, right $=-81.8$, and $-299<-81.8$ is true, so $0$ is in the solution set. At $x=-50$: left $=-449$, right $=-491.8$, and $-449<-491.8$ is false - consistent with $-50 < -41.77$ being outside the set.