Solution of the Compound Inequality

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

Solution:

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

Step 1: Multiply all terms by 𝑑 to eliminate the denominator: $a . d \leq b - c x \leq d . e$
Step 2: Isolate the term with 𝑥 by subtracting 𝑏 from all parts: $a . d - b \leq - c x \leq d . e - b$
Step 3: Divide by −𝑐 to isolate 𝑥, remembering to reverse the inequality signs: $\frac{a . d - b}{- c} \geq x \geq \frac{d . e - b}{- c}$
Final Step: Simplify the expressions and write the solution in interval notation: $[m i n (\frac{a . d - b}{- c}, \frac{d . e - b}{- c}), m a x (\frac{a . d - b}{- c}, \frac{d . e - b}{- c})]$

Example:

Solve the compound inequality:

- 5 \leq (2 - 7 x) / 8 \leq 15

Step 1: Multiply all parts by 8: $- 5 * 8 \leq 2 - 7 x \leq 15 * 8$ .

Step 2: Subtract 2 from all parts: $(- 40 - 2) \leq - 7 x \leq (120 - 2)$ .

Step 3: Divide by -7 and reverse inequalities: $6.00 \geq x \geq - 16.86$ .

Solution set:

[- 16.86, 6.00]