Inequalities 3 — Heart of Algebra

Questions in this quiz (10)

1. A bakery produces three types of bread: rye, sourdough, and whole wheat. Each loaf of rye requires $$0.25$$ kg of flour and $$0.15$$ hours of labor. Each loaf of sourdough requires $$0.35$$ kg of flour and $$0.25$$ hours of labor. Each loaf of whole wheat bread requires $$0.20$$ kg of flour and $$0.10$$ hours of labor. The bakery has at most $$45$$ kg of flour and at most $$25$$ labor hours available each day. Due to demand, the number of sourdough loaves produced must be no more than twice the number of rye loaves produced. Let $$r$$ represent the number of rye loaves and $$s$$ represent the number of sourdough loaves. Which system of inequalities best represents this situation?

   • $$0.25r+0.35s\le45$$$$0.15r+0.25s\le25$$$$s\le2r$$
   • $$0.25r+0.35s\ge45$$$$0.15r+0.25s\le25$$$$s\le2r$$
   • $$0.25r+0.35s\le45$$$$0.15r+0.25s\ge25$$$$s\ge2r$$
   • $$0.25r+0.35s\le45$$$$0.15r+0.25s\le25$$$$s\ge2r$$

2. A manufacturing company produces two types of electronic devices, Device X and Device Y. Each Device X requires $$3.5$$ hours for assembly and $$2$$ hours for testing. Each Device Y requires $$4$$ hours for assembly and $$1.5$$ hours for testing. The company has at most $$220$$ assembly hours and at most $$110$$ testing hours available each week. Due to demand, the company must produce at least $$40$$ total devices each week. Additionally, the number of Device X produced must be no more than twice the number of Device Y produced. Let $$x$$ represent the number of Device X and $$y$$ represent the number of Device Y. Which system of inequalities best represents this situation?

   • $$3.5x+4y\le220$$$$2x+1.5y\le110$$$$x+y\ge40$$$$x\le2y$$
   • $$3.5x+4y\ge220$$$$2x+1.5y\le110$$$$x+y\ge40$$$$x\le2y$$
   • $$3.5x+4y\le220$$$$2x+1.5y$$ > $$110$$$$x+y$$ < $$40$$$$x\ge2y$$
   • $$3.5x+4y\le220$$$$2x+1.5y\le110$$$$x+y\ge40$$$$y\le2x$$

3. A factory produces two types of items, Type A and Type B. Producing one Type A item requires $$4$$ hours of machine time and $$3$$ kg of raw material. Producing one Type B item requires $$5$$ hours of machine time and $$4$$ kg of raw material. The factory has at most $$300$$ machine hours and $$220$$ kg of raw material available per day. Additionally, due to demand, the factory must produce at least $$25$$ Type A items and at least $$15$$ Type B items each day. If $$x$$ represents the number of Type A items and $$y$$ represents the number of Type B items produced each day, what is the maximum possible number of Type A items that can be produced per day?

   • $$45$$
   • $$53$$
   • $$55$$
   • $$60$$

4. A manufacturing company produces two types of electronic components, Component A and Component B. Each Component A requires $$5$$ hours of assembly time and $$4$$ units of a special material. Each Component B requires $$3$$ hours of assembly time and $$6$$ units of the special material. The company has at most $$280$$ hours of assembly time and at most $$320$$ units of special material available each week. Let $$x$$ represent the number of Component A produced and $$y$$ represent the number of Component B produced. Both $$x$$ and $$y$$ must be non-negative integers. If the company produces at least $$35$$ Component A, which of the following could be the number of Component B produced?

   • $$42$$
   • $$35$$
   • $$30$$
   • $$48$$

5. $$y\ge\frac{3}{4}x+5$$$$y\le-4x+24$$$$x\ge0$$The system of inequalities above is graphed in the $$xy$$-plane. If the point $$(x,y)$$ is a solution to the system of inequalities, what is the maximum possible integer value of $$x$$?

   • $$3$$
   • $$4$$
   • $$5$$
   • $$6$$

6. $$-5$$ < $$x$$ < $$8$$$$3$$ < $$y$$ < $$9$$Given the inequalities above, which of the following represents the possible range of values for the expression $$4x-3y?$$

   • $$-47$$ < $$4x-3y$$ < $$23$$
   • $$-43$$ < $$4x-3y$$ < $$29$$
   • $$-29$$ < $$4x-3y$$ < $$23$$
   • $$-23$$ < $$4x-3y$$ < $$29$$

7. A coffee shop sells three types of beverages: lattes, cappuccinos, and iced coffees. A latte costs $$5.50$$ and a cappuccino costs $$4.00$$. A customer has a budget of at most $$85$$. The customer wants to buy at least $$4$$ lattes, and the number of cappuccinos purchased must be at least two more than twice the number of lattes purchased. What is the maximum number of cappuccinos the customer can purchase?

   • $$14$$
   • $$15$$
   • $$16$$
   • $$18$$

8. A point $$(x,y)$$ is a solution to the system of inequalities: $$y>2x+7$$ $$y<-3x+22$$ Which of the following describes all possible values of $$y$$ for points $$(x,y)$$ that are solutions to the given system?

   • $$y$$ > $$13$$
   • $$y$$ < $$13$$
   • $$7$$ < $$y$$ < $$22$$
   • $$y$$ < $$7$$

9. $$3y-2x$$ > $$15$$$$4x+6$$ < $$18$$The point $$(x,y)$$ is a solution to the system of inequalities in the $$xy$$-plane. Which of the following describes the range of possible values for $$y$$?

   • $$y$$ > $$5$$
   • $$y$$ > $$6$$
   • $$y$$ > $$7$$
   • $$y$$ > $$8$$

10. The shaded region in the $$xy$$-plane represents the solution set to a system of inequalities. Which of the following systems of inequalities best represents this shaded region?

    • $$2x+3y\le12,\ x-2y\le4,\ x\ge0,\ y\ge0$$
    • $$2x+3y\ge12,\ x-2y\le4,\ x\ge0,\ y\ge0$$
    • $$2x+3y\le12,\ x-2y\ge4,\ x\ge0,\ y\ge0$$
    • $$2x+3y\le12,\ x+2y\le4,\ x\ge0,\ y\ge0$$