Solve the inequality (75% + 85%(2000 - x))/2000 >= 83% for x.

Step-by-step solution

1. 1

   Convert every percentage to a decimal first. Mixing $\%$ symbols with algebra is the fastest route to an error, so replace them once and for all: $75\%=0.75$, $85\%=0.85$, $83\%=0.83$.

   $\frac{0.75+0.85(2000-x)}{2000}\ \ge\ 0.83$

2. 2

   Expand the bracket in the numerator.

   $0.85(2000-x)=1700-0.85x\;\Longrightarrow\;\frac{1700.75-0.85x}{2000}\ \ge\ 0.83$

3. 3

   Clear the denominator. Multiplying both sides by $2000$ is safe without flipping the sign, because $2000>0$:

   $1700.75-0.85x\ \ge\ 1660$

4. 4

   Isolate the $x$ term. Subtracting $1700.75$ from both sides:

   $-0.85x\ \ge\ -40.75$

5. 5

   Divide by $-0.85$ and flip the inequality. This is the one step where the direction changes, because dividing by a negative number reverses the order:

   $x\ \le\ \frac{-40.75}{-0.85}=\frac{40.75}{0.85}=47.941176\ldots$

6. 6

   Check a value on each side. At $x=47$: $\tfrac{1700.75-39.95}{2000}=0.83040\ge0.83$ ✓. At $x=48$: $\tfrac{1700.75-40.8}{2000}=0.829975<0.83$ ✗ - so the boundary really does sit between $47$ and $48$, and if $x$ must be a whole number the answer is $x\le47$.