System of Inequalities — Heart of Algebra

Questions in this quiz (5)

1. The following system of inequalities is graphed in the $$xy$$-plane:$$y\ge2x+1$$$$y\ge-x-2$$Which quadrants contain solutions to the system?

   • Quadrants I and II
   • Quadrants I and IV
   • Quadrants II and III
   • Quadrants I, II, and III

2. Biologists have determined that for a lake ecosystem to remain stable, the number of fish $$f$$ must be at least twice the number of birds $$b$$. In addition, the total number of fish and birds in the lake must be at least $$300$$. Which of the following systems of inequalities represents these conditions?

   • $$f\ge2b$$ $$f-b\ge300$$
   • $$f\ge2b$$ $$f+b\ge300$$
   • $$f\le2b$$ $$f+b\le300$$
   • $$f$$ < $$2b$$ $$f-b\le300$$

3. Maria works as a personal trainer and limits her total training time to no more than $$25$$ hours per week. She can run a cardio session in $$45$$ minutes and a strength session in $$30$$ minutes. Each cardio session earns her $$30$$ dollars, and each strength session earns her $$45$$ dollars. She needs to earn at least $$1200$$ dollars per week to cover her expenses. If Maria completes $$c$$ cardio sessions and $$s$$ strength sessions in one week, which of the following systems of inequalities best represents her weekly time and earnings constraints?

   • $$\frac{3c}{4}+\frac{s}{2}\leq25$$ and $$30c+45s\geq1200$$
   • $$\frac{c}{2}+\frac{3s}{4}\leq25$$ and $$30c+45s\leq1200$$
   • $$45c+30s\leq25$$ and $$30c+45s\geq1200$$
   • $$\frac{3c}{2}+\frac{3s}{2}\leq25$$ and $$45c+30s\geq1200$$

4. Maria wants to build a rectangular dog park that has an area of at least 240 square meters and a perimeter of at least 60 meters. If the length of the park is $$l$$ meters and the width is $$w$$ meters, which of the following systems of inequalities represents Maria’s requirements?

   • $$lw\ge60$$ $$l+w\ge240$$
   • $$lw\ge240$$ $$l+w\ge60$$
   • $$lw\ge240$$ $$l+w\ge30$$
   • $$lw\ge120$$ $$l+w\ge60$$

5. $$y$$ > $$12x+c$$$$y$$< $$4x+d$$In the system of inequalities below, $$c$$ and $$d$$ are constants. If $$(2,30)$$ is a solution to the system, which of the following could be the value of $$d-c$$?

   • $$12$$
   • $$14$$
   • $$16$$
   • $$18$$