Solution of the Compound Inequality

Problem:
We are solving the compound inequality of the form: $$ a \le \frac{b - cx}{d} \le e $$

Solution:

We will follow the same approach as before by isolating π‘₯, applying the same property to all terms of the inequality, and then graphing the solution and writing it in interval notation.

  1. Step 1: Multiply all terms by 𝑑 to eliminate the denominator: $$ a . d \le b - cx \le d . e $$
  2. Step 2: Isolate the term with π‘₯ by subtracting 𝑏 from all parts: $$ a . d - b \le - cx \le d . e - b $$
  3. Step 3: Divide by βˆ’π‘ to isolate π‘₯, remembering to reverse the inequality signs: $$ \frac{ a . d - b}{- c} \ge x \ge \frac{d . e - b}{- c} $$
  4. Final Step: Simplify the expressions and write the solution in interval notation: $$ [min(\frac{ a . d - b}{- c}, \frac{d . e - b}{- c}), max(\frac{ a . d - b}{- c}, \frac{d . e - b}{- c}) ] $$

Example: