Question

1. gift certificates you received a $100 gift certificate to a clothing store. the store sells t - shirts for $15 and dress shirts for $22. you want to spend no more than the amount of the gift certificate. you want to leave at most $10 of the gift certificate unspent. you need at least one dress shirt. what are all of the possible combinations of t - shirts and dress shirts you could buy? 32. a. geometry graph the system of linear inequalities. b. describe the shape of the solution region. c. find the vertices of the solution region. d. find the area of the solution region. $xgeq2$, $ygeq - 3$, $x + yleq4$ 33. which region represents the solution of the system? $yleq-\frac{3}{2}x - 2$, $3y-9xgeq6$ (a) i (c) iii (b) ii (d) iv c open - ended write a system of linear inequalities with the given characteristic. 34. all solutions are in quadrant iii. 35. there are no solutions. 36. business a jeweler plans to produce a ring made of silver and gold. the price of gold is about $25 per gram. the price of silver is approximately $.40 per gram. she considers the following in deciding how much gold and silver to use in the ring. the total mass must be more than 10 g but less than 20 g. the ring must contain at least 2 g of gold. the total cost of the gold and silver must be less than $90. a. write and graph the inequalities that describe this situation. b. for one solution, find the mass of the ring and the cost of the gold and silver.

Explanation:

31.

Step1: Set up the inequalities

Let $x$ be the number of T - shirts and $y$ be the number of dress shirts. The cost constraint is $15x + 22y\geq90$ (since we want to spend at least $100 - 10=90$) and $15x + 22y\leq100$, and $y\geq1$, $x\geq0,y\geq0$ and $x,y$ are non - negative integers.

Step2: Consider the case when $y = 1$

If $y = 1$, then the inequality $15x+22\times1\geq90$ gives $15x\geq90 - 22=68$, so $x\geq\frac{68}{15}\approx4.53$, and $15x+22\times1\leq100$ gives $15x\leq100 - 22 = 78$, so $x\leq\frac{78}{15}=5.2$. Since $x$ is an integer, $x = 5$.

Step3: Consider the case when $y=2$

If $y = 2$, then $15x+22\times2\geq90$ gives $15x\geq90 - 44 = 46$, so $x\geq\frac{46}{15}\approx3.07$, and $15x+22\times2\leq100$ gives $15x\leq100 - 44=56$, so $x\leq\frac{56}{15}\approx3.73$. Since $x$ is an integer, $x = 3$.

Step4: Consider the case when $y = 3$

If $y=3$, then $15x + 22\times3\geq90$ gives $15x\geq90-66 = 24$, so $x\geq\frac{24}{15}=1.6$, and $15x+22\times3\leq100$ gives $15x\leq100 - 66 = 34$, so $x\leq\frac{34}{15}\approx2.27$. Since $x$ is an integer, $x = 2$.

Step5: Consider the case when $y=4$

If $y = 4$, then $15x+22\times4\geq90$ gives $15x\geq90 - 88 = 2$, so $x\geq\frac{2}{15}\approx0.13$, and $15x+22\times4\leq100$ gives $15x\leq100 - 88 = 12$, so $x\leq\frac{12}{15}=0.8$. Since $x$ is an integer, $x = 0$.

The possible combinations are $(5,1),(3,2),(2,3),(0,4)$.

32.

Step1: Graph the inequalities

For $x\geq2$, it is the region to the right of the vertical line $x = 2$. For $y\geq - 3$, it is the region above the horizontal line $y=-3$. For $x + y\leq4$, rewrite it as $y=-x + 4$, and it is the region below the line $y=-x + 4$.

Step2: Describe the shape

The solution region is a triangle.

Step3: Find the vertices

Intersection of $x = 2$ and $y=-3$ is $(2,-3)$. Intersection of $x = 2$ and $x + y=4$ (substitute $x = 2$ into $x + y=4$ gives $y = 2$) is $(2,2)$. Intersection of $y=-3$ and $x + y=4$ (substitute $y=-3$ into $x + y=4$ gives $x = 7$) is $(7,-3)$.

Step4: Calculate the area

The base of the triangle is $7 - 2=5$ and the height is $2-(-3)=5$. The area $A=\frac{1}{2}\times base\times height=\frac{1}{2}\times5\times5=\frac{25}{2}=12.5$.

Answer:

1. The possible combinations are $(5,1),(3,2),(2,3),(0,4)$.
2. a. Graph the regions as described above. b. The shape is a triangle. c. The vertices are $(2,-3),(2,2),(7,-3)$. d. The area is $12.5$.