Tamara has $\$ 30,000$, part or all of which she wants to invest into a combination of corporate bonds and municipal bonds. She wants to invest no less than $\$ 8,000$ into corporate bonds, and at least three times as much into corporate bonds than into municipal bonds. Let $\mathrm{x}$ be the amount invested in corporate bonds, and let $\mathrm{y}$ be the amount invested in municipal bonds. Which system of inequalities describes Tamara's investment options? \[ \begin{array}{l} x+y \leq 30,000 \\ x \leq 8,000 \\ x \geq 3 y \end{array} \] \[ \begin{array}{l} x+y \leq 30,000 \\ x \geq 8,000 \\ x \geq 3 y \end{array} \] \[ \begin{array}{l} x+y \leq 30,000 \\ x \leq 8,000 \\ x \leq 3 y \end{array} \] \[ \begin{array}{l} x+y \leq 30,000 \\ x \geq 8,000 \\ x \leq 3 y \end{array} \]
Solution
Step 1 ：Let \( x \) be the amount invested in corporate bonds, and let \( y \) be the amount invested in municipal bonds.
Step 2 ：Tamara has a total of \( \$30,000 \) to invest, so the sum of \( x \) and \( y \) cannot exceed \( \$30,000 \). This gives the inequality \( x + y \leq 30,000 \).
Step 3 ：Tamara wants to invest no less than \( \$8,000 \) into corporate bonds, so \( x \) must be greater than or equal to \( \$8,000 \). This gives the inequality \( x \geq 8,000 \).
Step 4 ：Tamara wants to invest at least three times as much into corporate bonds than into municipal bonds, so \( x \) must be at least three times \( y \). This gives the inequality \( x \geq 3y \).
Step 5 ：The system of inequalities that describes Tamara's investment options is: \( \boxed{ \begin{array}{l} x+y \leq 30,000 \\ x \geq 8,000 \\ x \geq 3 y \end{array} } \)
