Angela is taking a math exam and needs to complete at least 10 geometry and algebra problems within 2 hours. Given Angela's proficiency in these two subjects, it will take her 30 minutes to complete the solution for a geometry problem, and 10 minutes to complete the solution for an algebra problem. Below, the variable $$ A $$ represents the number of algebra problems, and the variable $$ G $$ represents the number of geometry problems. Enter an appropriate value in each blank box below to complete the system of inequalities that define the constraints described above in the problem statement. Please Note: Where applicable, to indicate $$ \geq $$ enter $$ = $$, and to indicate $$ \leq $$ enter

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Angela is taking a math exam and needs to complete at least 10 geometry and algebra problems within 2 hours. Given Angela's proficiency in these two subjects, it will take her 30 minutes to complete the solution for a geometry problem, and 10 minutes to complete the solution for an algebra problem. Below, the variable $$ A $$ represents the number of algebra problems, and the variable $$ G $$ represents the number of geometry problems. Enter an appropriate value in each blank box below to complete the system of inequalities that define the constraints described above in the problem statement. Please Note: Where applicable, to indicate $$ \geq $$ enter $$ >= $$, and to indicate $$ \leq $$ enter <=

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To define the system of inequalities based on Angela's constraints, we need to consider the following: 1. **Time Constraint**: Angela has a total of 2 hours (which is 120 minutes) to complete the problems. The time taken for each type of problem is: - Geometry problem: 30 minutes - Algebra problem: 10 minutes Therefore, the total time spent on both types of problems can be expressed as: $$ 30G + 10A \leq 120 $$ 2. **Problem Count Constraint**: Angela needs to complete at least 10 problems in total, which can be expressed as: $$ G + A \geq 10 $$ 3. **Non-negativity Constraint**: Since the number of problems cannot be negative, we have: $$ G \geq 0 $$ $$ A \geq 0 $$ Now, we can summarize the system of inequalities: 1. $$ 30G + 10A \leq 120 $$ 2. $$ G + A \geq 10 $$ 3. $$ G \geq 0 $$ 4. $$ A \geq 0 $$ Thus, the completed system of inequalities is: - For the time constraint: $$ 30G + 10A <= 120 $$ - For the problem count constraint: $$ G + A >= 10 $$ - For non-negativity: $$ G >= 0 $$ and $$ A >= 0 $$ You can fill in the blanks as follows: 1. $$ 30G + 10A <= 120 $$ 2. $$ G + A >= 10 $$ 3. $$ G >= 0 $$ 4. $$ A >= 0 $$ $$ 30G + 10A \leq 120 $$, $$ G + A \geq 10 $$, $$ G \geq 0 $$, $$ A \geq 0 $$

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