Question

12 valeria, a music therapist, charges $50 for a half - hour session and $90 for an hour session. her expenses are $250 per week. one half of her patients have hour sessions. write an inequality to show how many patients, p, valeria must see each week to make a gross profit of $1570 or more. 13 solve the inequality.

Explanation:

Step1: Define revenue per patient type

Half of $p$ patients have half-hour sessions ($\$50$ each), half have hour sessions ($\$90$ each):
Revenue = $\frac{p}{2} \times 50 + \frac{p}{2} \times 90$

Step2: Set up profit inequality

Profit = Revenue - Expenses, profit ≥ $\$1570$, expenses = $\$250$:
$$\frac{p}{2} \times 50 + \frac{p}{2} \times 90 - 250 \geq 1570$$

Step3: Simplify revenue terms

Combine the patient revenue:
$$25p + 45p - 250 \geq 1570$$
$$70p - 250 \geq 1570$$

Step4: Isolate the variable term

Add 250 to both sides:
$$70p \geq 1570 + 250$$
$$70p \geq 1820$$

Step5: Solve for $p$

Divide both sides by 70:
$$p \geq \frac{1820}{70}$$
$$p \geq 26$$

Answer:

For question 12:

$\frac{p}{2} \times 50 + \frac{p}{2} \times 90 - 250 \geq 1570$ (or simplified: $70p - 250 \geq 1570$)

For question 13:

$p \geq 26$