Question: 3 6 Solve Compound Inequalities
LEARNING OBJECTIVE: Solve a compound inequality.
The value of a duplicating machine after $x$ copies have been made is given by
$V = 3,500 - 0.04x$.
After a few years, an appraiser states that the value of the machine is between
2,000 and 3,000, which can be represented by the following inequality:
$$2000 \leq 3500 - 0.04x \leq 3000$$
Which of the following inequalities represents the range of the number of
copies that were made?
a.) $2500 \leq x \leq 3750$
b.) $50000 \leq x \leq 75000$
c.) $12500 \leq x \leq 37500$
d.) $5000 \leq x \leq 7500$
1. **State the problem:** We need to find the range of $x$ (number of copies) such that the value $V$ of the machine satisfies $$2000 \leq 3500 - 0.04x \leq 3000.$$
2. **Write the compound inequality:** $$2000 \leq 3500 - 0.04x \leq 3000.$$
3. **Solve the left inequality:** $$2000 \leq 3500 - 0.04x.$$
Subtract 3500 from both sides:
$$2000 - 3500 \leq -0.04x$$
$$-1500 \leq -0.04x.$$
Divide both sides by $-0.04$ and reverse the inequality sign because dividing by a negative number:
$$\frac{-1500}{\cancel{-0.04}} \geq \frac{-0.04x}{\cancel{-0.04}}$$
$$37500 \geq x$$
or equivalently
$$x \leq 37500.$$
4. **Solve the right inequality:** $$3500 - 0.04x \leq 3000.$$
Subtract 3500 from both sides:
$$-0.04x \leq 3000 - 3500$$
$$-0.04x \leq -500.$$
Divide both sides by $-0.04$ and reverse the inequality sign:
$$\frac{-0.04x}{\cancel{-0.04}} \geq \frac{-500}{\cancel{-0.04}}$$
$$x \geq 12500.$$
5. **Combine the two results:**
$$12500 \leq x \leq 37500.$$
6. **Interpretation:** The number of copies made, $x$, is between 12,500 and 37,500 inclusive.
7. **Answer choice:** This corresponds to option (c)
**Final answer:** $$12500 \leq x \leq 37500.$$